a

Cycle Polytopes and their Application in Coding Theory

Sebastian Heupel

Juli, 2009 Examensarbeit

Department of Mathematics Technische Universit¨at Kaiserslautern Germany

Contents 1 Introduction

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2 Coding Theory 2.1 The basic model of Coding Theory . . . . . . . . . . . . . . . 2.2 Binary Linear Codes . . . . . . . . . . . . . . . . . . . . . . . 2.3 ML Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Matroids 3.1 Basic Definitions . . . . . . . . . . . 3.2 Examples for Matroids . . . . . . . . 3.2.1 Uniform Matroids . . . . . . . 3.2.2 The Cycle Matroid of a Graph 3.3 Matroid Duality . . . . . . . . . . . . 3.4 Representable Matroids . . . . . . . . 3.5 Graphic and Cographic Matroids . . 3.6 Circuit Space in Binary Matroids . . 3.7 Submatroids . . . . . . . . . . . . . . 3.8 Excluded Minor Theorems . . . . . .

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4 Composition and Decomposition of Matroids 4.1 Sums of Matroids . . . . . . . . . . . . . . . . . . . . . . 4.2 Construction of 2-Sum (De)Compositions . . . . . . . . . 4.3 Construction of 3-Sum (De)Compositions . . . . . . . . . 4.3.1 Construction of ∆-Sum (De)Compositions . . . . 4.3.2 Construction of Y -Sum (De)Compositions . . . . 4.4 Graphical Interpretation of Direct Sum (De)Composition

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1 1 2 5

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7 7 10 10 11 14 16 19 24 29 34

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39 39 46 48 49 51 54

CONTENTS 4.5 4.6

4.7 4.8

ii

Graphical Interpretation of 2-Sum (De)Composition . . . . . Graphical Interpretation of 3-Sum (De)Composition . . . . . 4.6.1 Graphical Interpretation of ∆-Sum (De)Composition 4.6.2 Graphical Interpretation of Y -Sum (De)Composition Properties of k-Sums . . . . . . . . . . . . . . . . . . . . . . (De)Composition Theorems . . . . . . . . . . . . . . . . . .

5 Cycle Polytope of Binary Matroids 5.1 Polyhedral Theory . . . . . . . . . . . . . . 5.2 Cycle Polytopes . . . . . . . . . . . . . . . . 5.3 Master Polytopes . . . . . . . . . . . . . . . 5.4 Lifting facets of Master Polytopes . . . . . . 5.5 Cycle Polytopes of Binary Uniform Matroids 5.6 Cycle Problem of Binary Matroids . . . . . . 6 Codes and their relation to Matroids 6.1 Binary Matroids and Binary Linear Codes . 6.2 Composition and Decomposition of Codes . 6.2.1 Direct Sum . . . . . . . . . . . . . . 6.2.2 2- and 3-Sum . . . . . . . . . . . . . 6.2.3 Equivalent 2- and 3-Sums . . . . . . 6.3 ML Decoding . . . . . . . . . . . . . . . . . 6.4 Hamming Codes and Dual Hamming Codes

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107 . 107 . 111 . 112 . 112 . 114 . 122 . 126

7 Conclusion 131 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Bibliography

135

List of Figures

139

Declaration

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Chapter 1 Introduction In this thesis, relations between binary matroids and binary linear codes are studied. In particular, cycle polytopes as well as decomposition of matroids yield interesting results for ML-Decoding. The term matroid was introduced in 1935 in H. Whitney’s paper [25] “On the abstract properties of linear dependence”. A matroid is a structure that was developed from the idea of abstract independence. Most frequently, a matroid is defined by means of independent sets. But there are many different ways of characterizing it. One of these definitions uses circuits. This approach to matroids is the most logical in the context of this work since circuits offer a possibility to relate binary matroids to binary linear codes. Polytopes associated with binary matroids were investigated by Barahona and Gr¨otschel in 1986 in their work “On the Cycle Polytope of a Binary Matroid” [2]. One more important contribution to the same topic is the paper “Master Polytopes for Cycles of Binary Matroids” [11] published by Gr¨otschel and Truemper in 1989. In the work “Decomposition and Optimization over Cycles in Binary Matroids” [10] also published by Gr¨otschel and Truemper in 1989, the cycle problem of binary matroids is studied. The results are developed from the investigations of cycle polytopes as well as from matroid decomposition introduced by Seymour in 1980 in his famous work “Decomposition of Regular Matroids” [20]. In the basic model of Coding Theory, codewords are transmitted over a chaniii

iv nel to a receiver. Because of the “noise” in the channel the receiver might get a corrupt word. Therefore, the receiving party must try to recover the original information from the received word. This process is called decoding. A particular approach to the problem of recovering the original codeword is searching for the codeword that maximizes the likelihood of the received word from the channel given that a particular codeword was sent. In Coding Theory, this approach is called ML-Decoding. Feldman showed in his PhD thesis “Decoding Error-Correcting Codes via Linear Programming” [6] that for common channels ML-Decoding is equivalent to the minimization of a linear function over all codewords. Investigating the connection between binary linear codes and binary matroids Kashyap noticed that ML-Decoding is equivalent to the cycle problem of binary matroids. In his paper “A Decomposition Theory for Binary Linear Codes” [15] which appeared last year, Kashyap transferred the results from Matroid Theory concerning decomposition, cycle polytopes and the cycle problem to binary linear codes. As a result he could characterize several classes of binary linear codes for which the problem of ML-Decoding is polynomially solvable. Unfortunately, these codes are not “good codes” for practical use. What fascinates on this work and where the motivation for this work comes from is that optimization offers the possibility to bring together Coding Theory and Matroid Theory and that a new application field for the abstract theory of matroids is developed. In Chapter 1, an introduction to Coding Theory is given and the problem of ML-Decoding is formulated. The aim of Chapter 2 is to give an introduction to the theory of matroids and to fix the terminology concerning matroids used throughout this thesis. Chapter 3 contains the results of Seymour as well as Gr¨otschel and Truemper concerning matroid (de)composition. The results are explained by means of examples for graphic matroids. In Chapter 4, the results on cycle polytopes and the cycle problem are pre-

v sented. In Chapter 5, the results from Matroid Theory are transferred to Coding Theory following the paper of Kashyap [15]. Thereafter, the problem of MLDecoding is solved for a real code. At the end of Chapter 5, Hamming Codes and Dual Hamming Codes are considered in the context of the question if parts of the main results of this work can be applied to them. At the beginning of every chapter the main literature is indicated. All definitions, propositions, lemmata and theorems without an explicit citation are closed to the indicated literature. Own results and own proofs are pointed out. Theorems and propositions which are directly taken from the literature have a citing remark. All examples presented in this thesis are own work. I would like to thank my supervisor Jun. Prof. Dr. Stefan Ruzika and Dipl.-Math. Akin Tanatmis for the initiation of this work, their support and contributions. I also would like to thank Prof. Dr. Horst W. Hamacher for his effort to read and evaluate my thesis.

vi

Chapter 2 Coding Theory This chapter is an introduction to the theory of codes. Besides an elementary part, Hamming Codes are introduced since they play an important role in the remainder of this thesis. In the last section, the problem of maximumlikelihood decoding (ML-Decoding) is formulated. This problem relates Coding Theory to Optimization since ML-Decoding can be written as a combinatorial optimization problem as it is shown in Chapter 6. The basic definitions given in this chapter are taken from the book [3] of Beutelspacher and from the PhD thesis of Feldman [6].

2.1

The basic model of Coding Theory

Coding Theory is based on the following communication model, see Figure 2.1. An information word x ∈ {0, 1}k is transformed into a codeword c ∈ {0, 1}n with n > k by an encoder which supplements redundant information to the initial word x. The codeword c is sent over a channel to a receiver. During the transmission of the codeword c several components of c, the so called code bits, might be changed. Thus, the received word c˜ is in general different from the codeword c that was sent. The errors occur with a certain probability that depends on the channel. In general, a channel is memoryless, i.e. each code bit is affected by the error probability independently. The transformation from c to c˜ can be described by the addition of an error vector e to c, i.e. c˜ = c + e. The decoder tries to identify and to correct the errors occurred 1

2.2. Binary Linear Codes

2

during the transmission. If the decoded word c¯ is equal to the codeword c the information word can be reconstructed and the decoder succeeds.

Figure 2.1: Basic Model of Coding Theory The challenge in the decoding process is to find the codeword c that was sent if c˜ is received. In practice, a decoder is actually an implementation of an algorithm. The question is often if the decoding process can be done in polynomial time for a given code and a given channel.

2.2

Binary Linear Codes

A binary code C is a subset of {0, 1}n . The code length n specifies the number of code bits in any codeword c ∈ C. To describe the relation of two given codewords the following is important. Definition 2.1. The Hamming distance d(c, c′ ) of two codewords c and c′ is defined to be the number of code bits in which c and c′ are different,i.e, d(c, c′ ) := |{i : ci 6= c′i }| .

2.2. Binary Linear Codes

3

An important property of a code is its minimum distance defined by dmin :=

min d(c, c′ ).

c,c′ ∈C;c6=c′

Next, a definition of a binary linear code is given. Definition 2.2. A code C is a binary linear code if it is a linear sub space of the vector space {0, 1}n . Thus any binary linear code can be described by a basis B = {b1 , . . . , bk } and, therefore, has dim(C) = k. A matrix G ∈ Mat(k × n, F2 ) where the rows are the basis vectors bj j = 1, . . . , k of a basis B of C is called a generator matrix of the code C. Thus, any codeword c can be written as c = xG where x ∈ {0, 1}k is an information word. Note that, in this work, all codewords and all information words are written as row vectors. Any binary linear code C has a dual code C ⊥ which is defined by C ⊥ := {d ∈ {0, 1}n : d · c = 0 ∀c ∈ C} Pn where d · c = i=1 ci di mod 2 is the inner product of the vector n space {0, 1} . Obviously, C ⊥ is equal to the null space of G. Hence, dim(C ⊥ ) = n − rank(G) = n − k and C and C ⊥ are orthogonal subspaces of {0, 1}n . A matrix H ∈ Mat((n − k) × n, F2 ) whose rows form a basis of the dual code C ⊥ is called a parity check matrix of C. Since c · d = 0 for all c ∈ C and d ∈ C ⊥ it follows that for all c ∈ C and any parity check matrix H of C cH T = 0. Thus, the code C is not only uniquely defined by any generator matrix G but also by any parity check matrix H. A parity check matrix H is said to have systematic structure if its of the

2.2. Binary Linear Codes

4

form H = (In−k |R) where In−k is an identity matrix of size n − k and R ∈ Mat((n−k)×k, F2 ). If H has systematic structure there exists a generator matrix G of C that is of the form G = (RT |Ik ) as it is shown in Chapter 3. An other property of a codeword c of a binary linear code is its weight w(c) which is defined to be the number of code bits that are different from zero. Thus, for any binary linear code C w(c) = d(c, 0) and w(c − c′ ) = d(c, c′ ) for all c, c′ ∈ C. Usually, in the literature, the abbreviation [n, k, d]−code stands for a binary linear code of code length n, dimension k and minimum distance d. Example 2.3. For any positive integer m ≥ 3, a Hamming Code is defined by a parity check matrix that has all 2m − 1 non zero binary vectors of length m as its columns. For m = 3, there are 7! different Hamming Codes of length 7. All parity check matrices of these codes have rank 3. Hence the dimension of all these Hamming Codes is k = 7 − 3 = 4. The [7, 4] Hamming Code that has   1 0 0 0 1 1 1 H = 0 1 0 1 0 1 1 0 0 1 1 1 0 1 as a parity check matrix is denoted by H7 in this work. A generator matrix of this code has as rows 4 linearly independent vectors which are orthogonal to any row of H. Since H has systematic structure a generator matrix of H7 is given by   0 1 1 1 0 0 0 1 0 1 0 1 0 0  G= 1 1 0 0 0 1 0 . 1 1 1 0 0 0 1

2.3. ML Decoding

5

All codewords of H7 are (0 (1 (0 (1 (1 (1 (0 (0

0 1 1 0 1 0 1 0

0 1 1 1 0 0 0 1

0 0 1 0 0 1 0 0

0 0 0 1 0 0 1 0

0 0 0 0 1 0 0 1

0), 1), 0), 0), 0), 1), 1), 1),

(1 (1 (0 (0 (1 (0 (0 (1

1 0 1 0 0 0 1 1

0 1 1 1 0 0 0 1

1 1 0 1 0 1 1 1

1 0 1 1 1 1 0 1

0 1 1 0 1 1 1 1

0), 0), 0), 1), 1), 0), 1), 1)

1 0 1 1

0 1 1 1

1 1 0 0

1 0 1 0

0 0), 1 0), 1 0), 0 1)

and all codewords of the dual code H7⊥ are (0 (1 (0 (0

2.3

0 0 1 0

0 0 0 1

0 0 1 1

0 1 0 1

0 1 1 0

0), 1), 1), 1),

(1 (1 (0 (1

ML Decoding

Maximum-likelihood Decoding (ML Decoding) is one of the methods to estimate the transmitted codeword given a received word. One aim of this work is to classify binary linear codes for which ML Decoding can be done in polynomial time. First, the ML codeword is defined. Definition 2.4. For a given channel, the ML codeword c∗ is the codeword that maximizes the probability that c˜ was received given that c was sent, i.e. c∗ := arg max P (˜ c was received|c was transmitted) c∈C

Note, that the ML codeword is not necessarily the original transmitted codeword. A decoder has the ML-certificate property if whenever the output of the decoder is a codeword it is guaranteed to be the ML codeword. In what follows , it is shown that ML-Decoding is equivalent to the minimization of a linear cost function over all codewords c ∈ C.

2.3. ML Decoding

6

For a received word c˜ the log-likelihood ratio γi of a code bit ci is given by   P (˜ ci |ci = 0) γi := ln , see [6]. P (˜ ci |ci = 1) From this definition, it follows that γi < 0 if ci is more likely to be 1 and γi > 0 if ci is more likely to be 0. Pn i=1 γi ci is referred to as the cost of a particular codeword c. Theorem 2.5 (Feldman, see [6]). For any binary-input memoryless channel, the codeword of minimum cost is the maximum-likelihood codeword. Proof : Since the channel is memoryless n Y

c∗ = arg max c∈C

!

P (˜ ci |ci )

i=1

−ln

= arg min c∈C

n Y

!

P (˜ ci |ci )

i=1

−

= arg min c∈C

n X

!

ln(P (˜ ci |ci ))

i=1

Pn

ln(P (˜ c1 |0)) to the right-hand side yields ! n X ln(P (˜ c1 |0)) − ln(P (˜ ci |ci )) c∗ = arg min

Adding the constant

i=1

c∈C

= arg min c∈C

= arg min c∈C

i=1

ln



γi c i

!

X

i:ci =1 n X i=1

P (˜ ci |0) P (˜ ci |1)

!

2

Chapter 3 Matroids The aim of this chapter is to give an introduction to the theory of matroids and to fix the terminology concerning matroids. The content of this chapter is referred to the books of Oxley [18] and Welsh [24]. An additional source is the PhD thesis of F. Bunke [4]. The main part of the notation is common to the notation in the literature.

3.1

Basic Definitions

In this section, a definition of a matroid is given and the basic structures that determine a matroid are introduced. ˆ = (N, I) consisting of a finite set N , Definition 3.1. A matroid is a pair M called ground set and a collection I of subsets of N , called independent sets, such that (1), (2) and (3) are satisfied. (1) ∅ ∈ I. (2) A ∈ I, B ⊆ A ⇒ B ∈ I. (3) A, B ∈ I, |A| > |B| ⇒ ∃ e ∈ A \ B such that B ∪ {e} ∈ I. For an arbitrary set A ∈ 2N the rank of A is given by the rank function r : 2N → N whose value is the cardinality of a maximal independent subset of A, i.e. r(A) := max{|X| : X ⊆ A, X ∈ I} for A ∈ 2N . The rank of the matroid is the rank of the corresponding ground set N and is often denoted 7

3.1. Basic Definitions

8

ˆ ). by r(M ˆ . The collection of A maximal independent subset of N is called a base of M ˆ is denoted by B(M ˆ ). all bases of M A subset of N not belonging to I is called dependent. Next, special independent sets are considered. ˆ = (N, I) is a minimal Definition 3.2. A circuit C ⊆ N in a matroid M ˆ is denoted by dependent subset of N . The collection of all circuits of M ˆ ). C(M From this definition it follows that a circuit has the following properties which can easily be verified. (1) If C is a circuit then r(C) = |C| − 1. (2) If C is a circuit then |C| ≤ r(N ) + 1. (3) The only matroid with no circuits has a single base N . (4) Every proper subset of a circuit is independent. ˆ the so called circuit axioms (C1) to (C3) Theorem 3.3. For a matroid M hold. ˆ ). (C1) ∅ ∈ / C(M (C2) If C1 and C2 are distinct circuits then C1 * C2 . (C3) If C1 and C2 are circuits and e ∈ C1 ∩ C2 then there exists a circuit C3 such that C3 ⊂ (C1 ∪ C2 ) \ {e}. ˆ ). Proof : (C1): Since ∅ ∈ I, ∅ can not belong to C(M (C2): follows from the fact that a circuit is a minimal dependent set. (C3): Assume that (C1 ∪ C2 ) \ {e} does not contain any circuit. Hence (C1 ∪ C2 ) \ {e} is an independent set. Choose f ∈ C2 \ C1 . Such f exists since C2 \ C1 6= ∅ by (C2). Since C2 is a circuit, C2 \ {f } is an independent set. Let X be a maximal independent subset of C1 ∪ C2 containing C2 \ {f }. Since C1 is a circuit, there exists some element g of C1 that is not in X.

3.1. Basic Definitions

9

Both f and g are no elements of X and they are distinct because f ∈ C2 \ C1 . Hence |X| ≤ |(C1 ∪ C2 ) \ {f, g}| = |C1 ∪ C2 | − 2 < |(C1 ∪ C2 ) \ {e}|. By 1.1.1 (3), there exists an element h of (C1 ∪ C2 ) \ ({e} ∪ X) =: Y such that Y ∪ {h} ∈ I but this is a contradiction to the maximality of X. 2 ˆ ) and the set of circuits C(M ˆ ) of a The set of independent sets I(M ˆ are related by I(M ˆ ) = {X ⊆ N : C * X ∀ C ∈ C(M ˆ )}. matroid M S ˆ) = ˆ Moreover, it holds I(M B∈B{X ⊆ B}, since any base of M is a maximal independent subset of N . Consequently, a matroid is uniquely determined by its set C of circuits or by its set B of bases. ˆ. By their definition, circuits are special independent sets of a matroid M Now, special circuits are introduced. ˆ if {e} is a circuit, i.e. An element e ∈ N is called a loop of the matroid M {e} is a dependent set. Two elements e, f ∈ N are parallel if they are not loops but {e, f } is a dependent set or, equivalently, if {e, f } is a circuit. ˆ , i.e. e and f are parallel It follows that if {e, f } is a circuit of a matroid M elements, then neither e nor f can be a loop. ˆ is a maximal subset X ⊆ N Definition 3.4. A parallel class of a matroid M with the property that no element of X is a loop, but any two distinct elements of X are parallel. A parallel class is called trivial if it consists of just one element. ˆ has no loops and no parallel elements it has only trivial parallel classes If M and is called simple. A circuit that consists of exactly three elements is called a triangle. ˆ = (N, I) be a matroid. Let I ∈ I be an independent Lemma 3.5. Let M set and let e ∈ N \ I. Then I ∪ {e} contains at most one circuit. Moreover, if B is a base and e ∈ N \ B then there exists a unique circuit C(B, e) such that e ∈ C(B, e) ⊆ B ∪ {e}. Proof : Suppose that I ∪ {e} contains the two distinct circuits C1 and C2 , i.e. C1 ∪ C2 ⊆ I ∪ {e}. Since I is independent, e must be contained in both C1 and C2 . By circuit axiom (C3) there exists a circuit C3 such that

3.2. Examples for Matroids

10

C3 ⊆ (C1 ∪ C2 ) \ {e} ⊆ I. But this contradicts the independence of I. Since B is a minimal spanning subset, the second statement follows directly. 2

Definition 3.6. The circuit C(B, e) from the above Lemma is called the fundamental circuit of e corresponding to the base B. All fundamental circuits that can be derived from the base B are collected in the set FB = {C(B, e) : e ∈ N \ B} which is called the fundamental circuit set corresponding to the base B. ˆ are components of another structure which The circuits of a matroid M is defined next. ˆ = (N, I) is a dependent set Definition 3.7. A cycle C ⊆ N of a matroid M that is the disjoint union of circuits or ∅.

3.2 3.2.1

Examples for Matroids Uniform Matroids

A first example for matroids can be given by the uniform matroids which form a simple, nevertheless important class of matroids. Definition 3.8. Let N be a finite set of cardinality n. Let 0 ≤ m ≤ n be an integer. The uniform matroid Um,n of rank m on N is defined as the matroid (N, I) with I(Um,n ) = {X ⊆ N : |X| ≤ m}. Clearly, the bases of Um,n are given by B(Um,n ) = {B ⊆ N : |B| = m} and the circuits by C(Um,n ) = {C ⊆ N : |C| = m + 1}. For X ⊆ N the rank function is given by: ( |F | if |F | ≤ m r(X) = m if |F | > m The matroids of the form Un,n have no dependent sets. If m = 0 every element of N is dependent and therefore a loop. In U1,n the set N = {1, 2, . . . , n} forms the single parallel class in U1,n . For m ≥ 2, Um,n has no loops and no

3.2. Examples for Matroids

11

parallel elements. In this case all parallel classes of Um,n contain just one element and the matroid is simple.

3.2.2

The Cycle Matroid of a Graph

Recall that a graph G = (V, E) is a pair consisting of two finite sets V containing the vertices and E containing unordered pairs of vertices called edges. If e = (u, v) ∈ E for u, v ∈ V then u and v are called incident. For a vertex v its degree d(v) indicates the number of edges incident to v. A cycle of G is a subgraph in which every vertex has degree d(v) = 2. A cycle in which all vertices are distinct is called simple. If every pair of vertices is joined by a sequence of distinct vertices and edges then a graph G is termed connected. A graph which is not connected is called disconnected Theorem and Definition 3.9. Let G = (V, E) be a graph and C ⊆ 2E be the set of edge sets of simple cycles of G. Then (E, C) is a matroid. This ˆ (G) and called the cycle matroid of matroid derived from G is denoted by M G. A proof of this statement can be found in [18], Chapter 1. From the definition of the cycle matroid, it follows that an edge set X ⊆ E is independent if and only if the edge set X does not form a cycle in the graph G. Consequently, if X ⊆ E is independent then G[X], the subgraph of G induced by X, is a forest of G, i.e. a not necessarily connected graph ˆ (G)) is the set of all forests of G. which does not contain a cycle. Hence I(M Since a spanning tree T of a graph G is a connected subgraph of G without ˆ (G), i.e. a maximal independent any cycle and V [T ] = V , a base B of M subset of E, is the edge set of a spanning tree of G. Example 3.10. Consider the complete graph K5 being displayed in Figure 3.1 where every pair of vertices is connected by an edge. So the number of edges of K5 corresponds to the number of possibilities to choose two from
five elements expressed by 52 and is equal to 10. The set E will be indicated by E = {1, 2, . . . , 10}, where, for example, 1 is a short cut for the edge e1 = (1, 5) etc.

3.2. Examples for Matroids

12

Figure 3.1: Complete Graph K5 All subsets of E of cardinality less or equal to two are independent and ˆ (K5 )). From the 120 subsets of E of cardinality therefore elements of I(M ˆ (K5 )) which do not form a cycle of K5 . The set three belong those to I(M ˆ (K5 )) of bases of the cycle matroid of K5 contains all edge sets of carB(M dinality four which do not form a cycle, i.e. the edge sets of the spanning ˆ (K5 )) is also contained in I(M ˆ (K5 )). trees of K5 . B(M All subsets of E of cardinality more than four are dependent, i.e. they contain at least one simple cycle of K5 . ˆ (K5 ) is connected since every two distinct vertices of K5 are conClearly, M tained in a cycle of the graph. The set B = {1, 2, 3, 10} is the edge set of a spanning tree of K5 and hence ˆ (K5 ). For every e ∈ N \ B there exists a unique fundamena base of M tal circuit corresponding to the base B. For e = 4 the fundamental circuit corresponding to B is equal to the edge set {2, 3, 4} = C(B, 4). Obviously, 4 ∈ C(B, 4) ⊆ B ∪ {4}. ˆ (K5 ) which is the disjoint The subset {1, 4, 5, 6, 7, 8} of E forms a cycle of M union of the two circuits {1, 4, 7} and {5, 6, 8}. Remark 3.11. Note, for no confusion, the edge set of every cycle in the

3.2. Examples for Matroids

13

ˆ (G). underlying graph G forms a circuit in the associated cycle matroid M ˆ (G) is the edge set of an Eulerian subgraph of the graph G, Any cycle of M i.e. a subgraph where every vertex has even degree. ˆ (G) is the union of the edge If a graph G is not connected, a base of M sets of spanning trees of the connected components of G. Since the cardinality of any base of a matroid is equal to the rank of this ˆ (G)) = |V | − t, where t is the number of connected matroid, it follows r(M components of G, i.e. the maximal connected subgraphs of G. These two facts can be verified in the following example. Example 3.12. Consider the graph H illustrated in Figure 3.2 which is a composition of K5 and an additional graph of two vertices.

Figure 3.2: Example for a disconnected graph Recall that a loop of a graph is an edge of the form (v, v) connecting a vertex with itself and that two edges are parallel edges if they have common endpoints but are not loops. The set T = {1, 4, 5, 8, 11} is the edge set of a spanning tree of H. Hence ˆ (H) of the graph H. As H consists of two T is a base of the cycle matroid M

3.3. Matroid Duality

14

ˆ (H)) is equal to 7 − 2 = 5. connected components r(M It can be observed that in the cycle matroid of a graph G loops and parallel elements coincide with the loops and the parallel edges of the underlying graph G. Note, that the cycle matroid of a graph can remain the same while the underlying graph is changed. An example for this is the addition of isolated vertices to a graph.

3.3

Matroid Duality

From Linear Algebra, it is known that every vector space has an orthogonal complement. In this Section, a similar theory called duality is introduced for matroids. Theorem and Definition 3.13. If {Bi : I ∈ I} is the set of bases of a ˆ on N then {N \ Bi : i ∈ I} is the set of bases of a matroid Mˆ ∗ on matroid M N. ˆ. Mˆ ∗ is called the dual matroid of M It holds: ˆ. (1) (Mˆ ∗ )∗ =M ˆ if and only if N \ X is spanning in Mˆ ∗ . (2) X ⊆ N is independent in M ˆ. (3) X ⊆ N is independent in Mˆ ∗ if and only if N \ X is spanning in M ˆ if and only if x belongs to every base of Mˆ ∗ . (4) x ∈ N is a loop of M ˆ ) + r(Mˆ ∗ ) = |N |. (5) r(M Since no proof was contained in the literature an own one is proposed. Proof : (1) follows from the definition of the dual matroid. (2) X ∈ I ⇔ ∃ base B such that X ⊆ B ⇔ N \ B ⊆ N \ X ⇔ N \ X is spanning in Mˆ ∗ . The argument in the last step is that N \ B is a base of Mˆ ∗ .

3.3. Matroid Duality

15

(3) apply (2) to Mˆ ∗ ˆ . So it must belong (4) As x is a loop, it is not contained in any base of M to every base of Mˆ ∗ and vice versa. (5) From the definition of the dual matroid it follows that the rank of Mˆ ∗ ˆ ). is equal to |N | − r(M 2 ˆ with M ˆ =Mˆ ∗ is called identically self dual. A matroid M ˆ 1 = (N1 , I1 ) and M ˆ 2 = (N2 , I2 ) are said to Definition 3.14. Two matroids M ˆ1 ∼ ˆ 2 , if there exists a one-to-one correspondence be isomorphic, written M =M between the elements of N1 and N2 that preserves independence. ˆ that is isomorphic to its dual Mˆ ∗ is said to be selfdual. The A matroid M ˆ are dualized by the following. A cobase of M ˆ is concepts of a matroid M ˆ is a circuit of Mˆ ∗ , two parallel elements in a base of Mˆ ∗ , a cocircuit of M ˆ and so on. Mˆ ∗ are called coparallel elements of M A cocircuit consisting of exactly three elements is called a triad. For the dual matroid Mˆ ∗ = (N, I)∗ the rank function r∗ : 2N → N is called the corank function and is defined by r∗ (A) := |A| + r(N \ A) − r(N ) for ˆ. A ∈ 2N where r is the rank function of M Definition 3.15. The cocycle matroid of a graph G is the dual of the cycle ˆ (G)∗ . matroid of the graph G and is denoted by M The bases of a cycle matroid are the edge sets of the spanning trees in the underlying graph. Likewise, the bases of the cocycle matroid are the edge sets of the complements of the spanning trees. A natural question to ask is now: How do the circuits of the cocycle matroid look like? For a graph G = (V, E) the set E \ T , where T is the edge set of a spanning tree of G, is a base of the cocycle matroid of the graph G. Hence the set (E \ T ) ∪ {t} for an arbitrary t ∈ T is a spanning set in Mˆ ∗ (G). The removal of the edges contained in this set from the graph G disconnects G and the resulting graph has one connected component more than G. So the considered spanning set (E \ T ) ∪ {t} is the edge set of a cut of G. Note that the set (E \ T ) ∪ {t} has not to be a circuit of Mˆ ∗ (G).

3.4. Representable Matroids

16

The following proposition states that every circuit of the cocycle matroid of a graph G is the edge set of a minimal cut of G. Proposition 3.16. Let G = (V, E) be a graph and X ⊆ E. Then the following are equivalent: ˆ (G)∗ . (1) X is a circuit of M ˆ (G). (2) X is a cocircuit of M (3) X is the edge set of a minimal cut of G. A proof of this proposition can be found in [18], Chapter 2. Recall that the size of a cut is the cardinality of the edge set of the cut. ˆ (G) and two coparallel So a cut consisting of a single edge is a coloop of M elements form a cut of size two. ˆ (K5 ) from Example 3.10 is {1, 5, 6, 7}. A cocircuit of M

3.4

Representable Matroids

In what follows representable matroids are introduced and some of their properties are deduced. Proposition 3.17. Let F be a field and let A ∈ Mat(r × n, F) be a matrix. Let N = {1, . . . , n} be the set of column labels of A. Define I to be the set of subsets I of N for which the column vectors A.j , j ∈ I are linearly independent in the vector space F r . Then (N, I) is a matroid. A proof of this proposition can be found in [18], Chapter 1. Definition 3.18. The matroid derived from A as described above is called ˆ (A). the matrix matroid of A and denoted by M ˆ (A). The matrix A is said to be a representation matrix for the matroid M If, in addition, A has linearly independent rows it is called a minimal representation. A minimal representation matrix A of the form A = (Ir |R) is

3.4. Representable Matroids

17

ˆ (A). Such A is often also said called a standard representation matrix for M to have systematic structure. By the definition of the matrix matroid, it follows that a loop of a repˆ is the index of an all zero column in a representation resentable matroid M ˆ. matrix for M ˆ (A) derived from a matrix A over the Remark 3.19. A matrix matroid M field F remains unchanged if one of the following operations are performed on A. (1) Interchanging two rows. (2) Multiplying a row by a non-zero element of F . (3) Replacing a row by the sum of that row and another. (4) Deleting a zero row if it is not the only row. (5) Interchanging two columns together with their labels. (6) Multiplying a non-zero column by a non-zero element of F. (7) Replacing each entry of A by its image under some automorphism of F. Hence, the representation matrix of a matrix matroid is not unique. The operations (1)-(3) are called elementary row operations. ˆ being isomorphic to the matrix matroid of Definition 3.20. A matroid M a matrix A over a field F is defined to be F-representable. In that case, the ˆ. matrix A is called an F-representation for M ˆ being representable over some field F is called representable. A matroid M ˆ being representable over the field F2 is called binary. A matroid M ˆ being representable over the field F3 is called ternary. A matroid M ˆ being representable over every field is called regular. A matroid M It can be shown that any regular matroid has a representation matrix A being totally unimodular, i.e the determinant of every square submatrix of A is in {−1, 0, 1}. The following theorem introduces a relation between a representable matroid and its dual.

3.4. Representable Matroids

18

Theorem 3.21. Let A ∈ Mat(r × n, F) be a F-representation for a matroid ˆ having rank r. M Then the dual matroid Mˆ ∗ is also F-representable and any matrix B ∈ Mat((n − r) × n, F) such that the rows span the null space of A is an Frepresentation for Mˆ ∗ . Proof : Let B ∈ Mat((n − r) × n, F) with the property that the rows of B span the null space of A. Hence, Ax = 0 for x ∈ F n if and only if there exists some y ∈ F n−r such that x = yB. (*) Assume that the first r columns of A are linearly dependent. Then, there exists a vector x = (x1 , x2 , . . . , xr , 0, . . . , 0) ∈ F n , x 6= 0 fulfilling Ax = 0. By (*) this is equivalent to x = yB for some y ∈ F n−r , y 6= 0. Writing B = (B1 |B2 ) with B1 ∈ Mat((n − r) × r, F) and B2 ∈ Mat((n − r) × (n − r), F) together with the fact that the last n − r components of x are equal to 0 implies yB2 = 0. Thus B2 is not regular because y 6= 0 and therefore the last n − r columns of B are linearly dependent. ˆ (A) remains unNote that by interchanging two columns the matroid M changed. So it is now shown that a subset of r columns of A is linearly dependent if and only if the complementary set of n − r columns of B is linearly dependent. Linear dependence can now be replaced by linear independence without changing the statement. ˆ (A) if and only if So a set of r columns of A is independent, i.e. a base of M the set of the remaining n − r columns of B is independent, i.e. a base of ˆ (A)∗ =M ˆ (B). M 2 Theorem 3.21 has the following corollary that is very important for the practical use. Corollary 3.22. Let (Ir |R) ∈ Mat(r × n, F) be a F-representation for a maˆ having rank r. troid M Then the dual matroid Mˆ ∗ is also F-representable and the matrix (−RT |In−r ) ∈ Mat((n − r) × n, F) is a F-representation for Mˆ ∗ .

3.5. Graphic and Cographic Matroids

19

Proof : From linear algebra it is known that the orthogonal subspace of the row space of a matrix (Ir |R) ∈ Mat(r × n, F) is the row space of the matrix (−RT |In−r ) ∈ Mat((n − r) × n, F). So the claim follows together with Theorem 3.21.

3.5

Graphic and Cographic Matroids

In this section, special regular matroids, the so called graphic and cographic matroids, are considered. ˆ (G) of Definition 3.23. A matroid being isomorphic to the cycle matroid M a graph G is called graphic. By defining corresponding graphs, it can be verified that all matroids with 3 or less elements are graphic. The smallest non-graphic matroid is the uniform matroid U2,4 . Indeed, it is impossible to draw a graph that consists of four edges and where all edge sets of cardinality three are cycles. Example 3.24. Consider again the complete Graph K5 introduced in Example 3.10. Recall that the vertex-edge incidence matrix of a graph G = (V, E) is a matrix A ∈ Mat(|V | × |E| , F2 ) where the ith component of every column Ae for e ∈ E is equal to one if vertex i is an endpoint of the edge e and zero otherwise. For K5 from Example 3.10 the vertex-edge incidence matrix has the following form.   1 0 0 0 1 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0    A= 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0

ˆ (A) represented by the It can easily be verified that the matrix matroid M ˆ (K5 ) of the graph K5 . Hence, matrix A is isomorphic to the cycle matroid M ˆ (A) is graphic by definition and also binary by construction. By Remark M

3.5. Graphic and Cographic Matroids ˆ (A) is also represented by the 3.19 M  1 0 0 0  0 1 0 0 A′ =  0 0 1 0 0 0 0 1

20

matrix 1 1 0 0

1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1

 0 0  1 1

ˆ (A) can be dualized by Corollary 3.22 and a Since A′ is in standard form M ˆ (K5 )∗ is representation matrix for M   1 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0   1 0 0 1 0 0 1 0 0 0  B= 0 1 1 0 0 0 0 1 0 0   0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1

ˆ (K5 )∗ , Here again, one can verify that the set {1, 5, 6, 7} is a circuit of M i.e. a minimal cut of the graph K5 .

These considerations do not only hold for K5 . One can generalize that every matrix matroid represented by the vertex-edge incidence matrix of some graph is graphic. Note that the corresponding graph has not to be connected, i.e. there might exist two vertices which are not joined by a sequence of distinct vertices and edges. Considering the opposite direction, one can even say that every graphic matroid is isomorphic to the cycle matroid of some connected graph.(for a proof see, e.g., [18], Chapter 1) The following example should illustrate these two statements. Example 3.25. Consider the following graph G drawn in Figure 3.3 which is not connected.   1 0 0 0 0 1   . 0 1 1 G has the vertex-edge incidence matrix A =    0 1 0 1 0 0

ˆ (A) is graphic by the above argument. By Remark The matrix matroid M

3.5. Graphic and Cographic Matroids

21

Figure 3.3: A disconnected graph G 3.19 it follows that another representation  1 1  0 1 A′ =  1 0 0 0

ˆ (A) is for M  0 1  0 1

A′ can be considered as vertex-edge incidence matrix of the graph H shown in Figure 3.4 which is connected.

Figure 3.4: A connected graph H

3.5. Graphic and Cographic Matroids

22

The next theorem states that a graphic matroid is not only binary, but also regular. Theorem 3.26. A graphic matroid is representable over every field. A proof can be found in [18], Chapter 5. Oxley also shows that a repreˆ (G) of a corresponding graph G is totally unimodular. sentation matrix for M Hence the statements below ˆ is regular (i) M ˆ being totally unimodular (ii) There exists a representation matrix of M are indeed equivalent as mentioned earlier in this chapter. ˆ (G)∗ Definition 3.27. A matroid being isomorphic to the cocycle matroid M of a graph G = (V (G), E(G)) is called cographic. A cographic matroid can also be graphic. If that is the case there exist a connected graph H = (V (H), E(H)) with the following properties: (1) E(G) = E(H). ˆ (H)) + 1 = |E(G)| − r(M ˆ (G)) + 1. (2) |V (H)| = r(M (3) The edge set of any cycle in G is the edge set of a cut in H and the edge set of any cycle in H is the edge set of a cut in G. (1) to (3) follow from the properties of the cycle matroid and its dual. ˆ (B) represented by the matrix B from Example 3.24 is The matroid M ˆ (K5 )∗ and thus cographic. The following proposition shows isomorphic to M that this matroid is not isomorphic to the cycle matroid of a graph. ˆ (K5 )∗ is not graphic. Proposition 3.28. M ˆ (K5 )∗ is graphic. Proof : Assume that M ˆ (K5 )∗ ∼ ˆ (H) for some connected graph H. Then M =M ˆ (K5 )∗ has 10 edges and 7 vertices and therefore By the above properties M < 3. Hence H has a vertex of degree at an average vertex degree of 2|E(G)| |V (G)|

3.5. Graphic and Cographic Matroids

23

ˆ (K5 )∗ has a circuit of size one or two. But most two which implies that M this means that K5 has a cycle of length one or two. Contradiction. 2 The class of matroids with graphic and cographic members should now be characterized. ˆ (G) be the cycle matroid of a graph G. Then M ˆ (G)∗ Proposition 3.29. Let M is itself graphic if and only if the graph G is planar. Definition 3.30. A graph G is called planar if it is isomorphic to a graph G ′ being contained in the plane such that no two edges intersect. Proposition 3.29 was first proved by Tutte. A proof can be found in [18], Chapter 2. This section is finished by illustrating the classification of the matroids that have been considered so far.(Figure 3.5) A similar figure can be found in [18], Chapter 6.

Figure 3.5: Classification of Matroids

3.6. Circuit Space in Binary Matroids

3.6

24

Circuit Space in Binary Matroids

In this section, an operation is introduced that can be interpreted as a sum for circuits. This operation reveals several properties of binary matroids. Moreover, a definition is given that links circuits of binary matroids with vectors in Fn2 . These vectors form a subspace of Fn2 which is considered in detail. It is shown how a basis of this subspace can be found and how this subspace can be defined by a corresponding representation matrix. Definition 3.31. For two finite sets E and F the symmetric difference is defined by E∆F := (E \ F ) ∪ (F \ E) or, equivalently, E∆F := (E ∪ F ) \ (E ∩ F ). Next, a characterization of binary matroids is given. ˆ the following statements are equivalent: Theorem 3.32. For a matroid M ˆ is binary. (a) M (b) For every circuit C and every cocircuit D, |C ∩ D| is even. ˆ is (c) The symmetric difference of every collection of distinct circuits of M ˆ. the disjoint union of circuits of M (d) If C1 and C2 are distinct circuits then C1 ∆C2 contains a circuit. ˆ , C = ∆e∈C\B C(B, e), where (e) For every base B and every circuit C of M C(B, e) is the fundamental circuit of e corresponding to B. For a proof, see [24], Chapter 10. Definition 3.33. For a subset E ⊆ N the incidence vector xE ∈ Fn2 corresponding to E is defined by ( 1 if e ∈ E (xE )e := 0 if e ∈ N \ E With the help of this definition a link between the circuits of binary matroids and vectors in Fn2 can be introduced as follows.

3.6. Circuit Space in Binary Matroids

25

Definition 3.34. The sub-vector space of Fn2 that is generated by the inciˆ is called the circuit space dence vectors of the circuits of a binary matroid M ˆ and is denoted by C(M ˆ ). of M Remark 3.35. The symmetric difference of circuits corresponds to the adˆ ) can be considered dition of their incidence vectors in Fn2 . Therefore, C(M as to consist of the incidence vectors of all possible symmetric differences of ˆ . In other words, the circuit space is equal to the set of the circuits of M ˆ. incidence vectors of the cycles of M A basis for the circuit space can be defined as follows. Definition 3.36. A set of Cycles CB = {C1 , . . . , Cn−r } being minimal with the property that the corresponding incidence vectors span the circuit space ˆ ) of a binary matroid M ˆ is called a circuit basis of M ˆ . Consequently, C(M ˆ has a unique representation in CB as every cycle C in M C = α1 C1 ∆α2 C2 ∆ . . . ∆αn−r Cn−r , αi ∈ {0, 1}. The following proposition states that a set of fundamental circuits is a basis of the circuit space. ˆ is Proposition 3.37. A basis for the circuit space of a binary matroid M given by the set FB of fundamental circuits corresponding to a base B. A ˆ in the basis FB is given basis representation for an arbitrary cycle C of M by C = ∆e∈C\B C(B, e). Proof : Since FB = {C(B, e) : e ∈ N \ B} it follows that the incidence vectors of the elements of FB are linearly independent. By Theorem 3.32 ˆ is represented by C = ∆e∈C\B C(B, e). These two every circuit C of M arguments together imply that FB is a basis for the circuit space of a binary matroid. The representation for an arbitrary cycle is given by Theorem 3.32. The next proposition introduces a relation between the circuit space and a representation matrix. ˆ ) of a binary matroid M ˆ is the Proposition 3.38. The circuit space C(M ˆ . The null space of a binary representation matrix A ∈ Mat(r × n, F2 ) of M ˆ ) is n − r. dimension of C(M

3.6. Circuit Space in Binary Matroids

26

Proof : The matrix A ∈ Mat(r × n, F2 ) describes a linear mapping A : F2n → F2r . So, it has to be shown that a vector xC ∈ F2n belongs to the kernel of the mapping A if and only if xC is the incidence vector of a cycle ˆ. C ⊆ N of M “⇒”: Let xC ∈ F2n be the incidence vector of a subset C ⊆ N and additionally xC ∈ kern(A). By assumption AxC = 0 which is equivalent to P i∈C Ai = 0 and implies that the columns of A indexed by i ∈ C are linearly ˆ and there exists a circuit dependent. Hence C ⊆ N is a dependent set of M ∅ 6= C1 ⊆ C. For C1 , two cases have to be distinguished. (1) If C1 = C then C is a circuit and this direction of the statement is proved. P (2) If C1 ⊆ C then i∈C1 Ai = 0 and, since AxC = 0, it also holds P j∈C\C1 Aj = 0. It follows that C \ C1 is again a dependent set of ˆ and the cases (1) and (2) have to be distinguished once more. M Since |C| is finite the procedure will be repeated until C is decomposed as C = C1 ∪ C2 ∪ . . . ∪ Ck ∪ Xk , where Xk = C \ (C1 ∪ C2 ∪ . . . ∪ Ck ) and ˆ . Moreover, P Ci , i = 1, . . . , k are circuits of M i∈Xk Ai = 0 and no proper circuit can be extracted from Xk . Thus Xk is itself a circuit. Consequently, ˆ. C is the disjoint union of circuits and therefore a cycle of M ˆ . Since C is the disjoint union of finitely many, “⇐”: Let C be a cycle of M P say k, circuits its incidence vector xC can be written as xC = ki=1 xCi . Since A is a linear mapping xc ∈ kern(A) if xCi ∈ kern(A) for every i = 1, . . . , k. ˆ belongs to So, it rest to show that the incidence vector of every circuit of M kern(A). ˆ . Since C is a dependent set the set of columns of A Let C be a circuit of M corresponding to C is linearly dependent. Since C is a minimal dependent P set it must hold i∈C Ai = 0 which is equivalent to AxC = 0 and implies xC ∈ kern(A). 2

3.6. Circuit Space in Binary Matroids

27

Definition 3.39. A matrix C ∈ Mat(s × n, F2 ) whose rows are the incidence ˆ , i.e. all members C1 , . . . , Cs of the circuit vectors of all cycles of a matroid M ˆ , is called the cycle matrix of M ˆ. space of M ˆ is a sub-matrix of the A circuit basis matrix CC ∈ Mat((n − r) × n, F2 ) of M ˆ. cycle matrix whose rows correspond to a basis CB of the circuit space of M A combination of Theorem 3.21, Proposition 3.38 and Definition 3.39 yield the following theorem. Theorem 3.40. Let A ∈ Mat(r × n, F2 ) be a representation matrix of a ˆ of rank r. binary matroid M Then the dual matroid Mˆ ∗ is also binary and the cycle matrix or any circuit ˆ is a representation matrix for basis matrix B ∈ Mat((n − r) × n, F2 ) of M Mˆ ∗ . The following theorem shows how a representation of Mˆ ∗ can be derived ˆ and vice versa. from a standard representation of M ˆ has the standard representation matrix Theorem 3.41. If the matroid M (Ir |R) ∈ Mat(r × n, F2 ) then the matrix (−RT |In−r ) ∈ Mat((n − r) × n, F2 ) is a F2 -representation for the dual matroid Mˆ ∗ . Proof : Follows directly from Theorem 3.40 and Corollary 3.22.

2

Next, it is shown that a standard representation matrix can be derived from a set of fundamental circuits and vice versa. Proposition 3.42. The fundamental circuits of a binary matroid ˆ corresponding to a base B determine a unique standard representation M ˆ with respect to that base B. More precisely, for i = matrix (Ir |R) of M r + 1, . . . , n, column ei of R contains the incidence vector of C(B, ei ) \ {ei }. Proof : Let C(B, ei ) = {bi1 , bi2 , . . . , bik , ei }. Since C(B, ei ) is a minimal dependent set addition modulo 2 of all column vectors of (Ir |R) whose labels are in C(B, ei ) must yield the zero vector. In other words, column ei of R is the sum modulo 2 of the columns bi1 , bi2 , . . . , bik . From this, it follows that column ei represents xC(B,ei )\{ei } . 2

3.6. Circuit Space in Binary Matroids

28

ˆ = (N, I) and f ∈ B the Definition 3.43. For a base B of a matroid M fundamental cocircuit D(B, f ) of f corresponding to the base B is defined by D(B, f ) := CMˆ ∗ (B ∗ , f ), where CMˆ ∗ (B ∗ , f ) is the fundamental circuit of f corresponding to the base B ∗ = N \ B of the dual matroid Mˆ ∗ . The next statement about the cocircuit space follows from Theorem 3.41 and Proposition 3.42. Proposition 3.44. Let A = (Ir |R) ∈ Mat(r × n, F2 ) be a standard repreˆ of rank sentation matrix with respect to the base B of a binary matroid M ˆ , i.e. the rows of A are r. Then the rows of A span the cocircuit space of M ˆ corresponding to the base B, where the fundamental cocircuits D(B, f ) of M f ∈ B. If the representation matrix is not in standard form the next proposition is important. ˆ is the row Proposition 3.45. The cocircuit space of a binary matroid M ˆ. space of a representation matrix of M ˆ and Proof : Assume that r is the rank of the representation matrix of M that its first r columns are linearly independent. If necessary, the last can ˆ . Then, by deletions of zero rows be realized by reordering the elements of M and by elementary row operations the matrix can be converted in a matrix of the form (Ir |R) without changing its row space. By Proposition 3.44 the ˆ . Every rows of the new matrix form a set of fundamental cocircuits of M element of the row space of the considered matrix is also in the cocircuit ˆ since the fundamental cocircuits span the cocircuit space. By space of M Theorem 3.32 applied to Mˆ ∗ there exists a basis representation in this set ˆ . Hence the incidence vecof fundamental cocircuits for every cocircuit of M tor of every cocircuit is contained in the row space of the considered matrix.2

3.7. Submatroids

3.7

29

Submatroids

In this section it will be shown how the concepts of restriction and contraction of graphs can be extended to matroids. In both cases, a new matroid on a ˆ on N . subset T ⊆ N will be derived from a matroid M ˆ = (N, I) be a matroid and T ⊆ N . Theorem and Definition 3.46. Let M ˆ |T ) := {X : X ⊆ T, X ∈ I(M ˆ )}. Define I(M ˆ |T ) is the set of independent sets of a matroid on T. This matroid Then I(M ˆ |T and called the restriction of M ˆ to T . The rank function is denoted by M ˆ |T is the restriction of the rank function r of M ˆ to T . The following r|T of M ˆ |T : hold for M ˆ and X ⊆ T then X is dependent in M ˆ |T . (1) If X is dependent in M ˆ |T has rank r(T). (2) M ˆ |T has as its circuits all circuits of M ˆ which are contained in T . (3) M ˆ = (N, I) is representable over a field F and the matrix Remark 3.47. If M ˆ then it is obvious that for T ⊆ N it holds A is an F-representation for M ˆ (A)|T = M ˆ (A|T ), where A|T is the matrix obtained from A by deleting M all the columns whose labels are in N \ T . ˆ = (N, I) be a matroid and T ⊆ N . Theorem and Definition 3.48. Let M ˆ .T ) := {X ⊆ T : ∃ maximal independent subset Y ⊆ N \ T s.t. Define I(M ˆ )}. X ∪ Y ∈ I(M ˆ .T ) is the set of independent sets of a matroid on T. This matroid Then I(M ˆ .T and called the contraction of M ˆ to T . The rank function is denoted by M ˆ .T is defined by r.T (A) := r(A ∪ (N \ T )) − r(N \ T ) for A ⊆ T . The r.T of M ˆ .T is r(N ) − r(N \ T ). rank of the matroid M By the following theorem, it will turn out that contraction is the dual operation of restriction. Furthermore, this theorem shows how restriction and contraction are related. ˆ = (N, I) be a matroid and T ⊆ N . Then (1) and (2) Theorem 3.49. Let M hold.

3.7. Submatroids

30

ˆ |T )∗ =Mˆ ∗ .T . (1) (M ˆ .T )∗ =Mˆ ∗ |T . (2) (M The proposed proof is a more extensive revision of the proof given in the book of Welsh [24]. ˆ by Proof : First observe that (1) can be transformed into (2) by replacing M Mˆ ∗ and taking duals. So, it suffices to show (1). ∗ Let X be an arbitrary subset of T . Then the statement is true if r|T (T \X) = ∗ r.T (T \ X). ∗ r.T (T \ X) = r∗ ((T \ X) ∪ (N \ T )) − r∗ (N \ T )

= |N \ X| + r(N \ (N \ X)) − r(N ) − (|N \ T | + r(N \ (N \ T )) − r(N )) = |N \ X| + r(X) − |N \ T | − r(T ) = |N | − |X| + r(X) − |N | + |T | − r(T ) = |T \ X| + r(X) − r(T ) = |T \ X| + r|T (X) − r|T (T ) ∗ = |T \ X| + r|T (T \ (T \ X)) − r|T (T ) = r|T (T \ X)

.

2

ˆ (N, I) be a matroid. For two disjoint subsets S Proposition 3.50. Let M and T of N , it holds ˆ |S)|T =M ˆ |(S ∪ T ) = (M ˆ |T )|S. (1) (M ˆ .S).T =M ˆ .(S ∪ T ) = (M ˆ .T ).S. (2) (M ˆ |S).T = (M ˆ .T )|S. (3) (M Proof : (1) and (2) follow directly from the definition of restriction and contraction. To prove (3) it has to be shown that both matroids in (3) have the same rank functions.

3.7. Submatroids

31

Let X be an arbitrary subset of N \ (S ∪ T ). Define rMˆ |T (X) to be the rank ˆ restricted to T and define r ˆ (X) to be r contracted to T . function r of M M .T Note that in this notation rMˆ .T (X) is equal to r.T (X). Then r(Mˆ .T )|S (X) = r.T (X) = r(X ∪ T ) − r(T ) = rMˆ |S (X ∪ T ) − rMˆ |S (T ) = r(Mˆ |S).T (X) 2 From now on, to shorten the arguments, the parentheses in expressions ˆ |S).T may be dropped under the condition that S and T are such as (M disjoint. ˆ = (N, I) be a matroid and T ⊆ N . A matroid O ˆ Definition 3.51. Let M ˆ if O ˆ is obtained by a sequence of restrictions on T is called a minor of M ˆ. and contractions of M ˆ of M ˆ with O ˆ 6=M ˆ is called a proper minor of M ˆ. A minor O ˆ can be Note that any sequence of restrictions and contractions from M ˆ |S.T for two disjoint sets S and T where one of them might be written as M the empty set. ˆ (N, I) be a matroid and S, T ⊆ N . Proposition 3.52. Let M ˆ =M ˆ |S.T is a minor of M ˆ if and only if Oˆ∗ =Mˆ ∗ .S|T is a minor of Then O Mˆ ∗ . Since a proof for this proposition is not contained in the literature an own one is proposed. Proof : By use of Theorem 3.49, it holds ˆ ∗ = (M ˆ |S.T )∗ O ˆ |S)∗ |T = (M ˆ ∗ .S|T = O ˆ∗ =M 2

3.7. Submatroids

32

Definition 3.53. A family M of matroids is called minor closed if for each ˆ ∈ M every minor of M ˆ is in M, too. M The class of uniform matroids is minor closed. Indeed, from Theorem and Definition 3.46 as well as Theorem and Definition 3.48 it follows that for a subset T of cardinality t ( ( U0,t if n − t ≥ m U if t ≤ m t,t and Um,n .T ∼ Um,n |T ∼ = = Um−n+t,t if n − t < m Um,t if t > m Thus any minor of a uniform matroid is again a uniform matroid. In the following it is shown that graphic matroids, F-representable matroids and regular matroids also form minor closed families. Proposition 3.54. Every minor of a graphic matroid is again a graphic matroid. Proof : It suffices to show that for every graph G = (V, E) and T ⊆ E ˆ (G|T ) = M ˆ (G)|T M

(3.7.1)

where G|T denotes the graph obtained from G by deleting the edges in E \ T and ˆ (G.T ) = M ˆ (G).T M (3.7.2) where G|T denotes the graph obtained from G by deleting the edges in E \ T and identifying their ends. (3.7.1) follows directly from the definition of restriction. (3.7.2) can be proved by an induction argument if one can show that ˆ (G).e) = I(M ˆ (G.e)) holds for all e ∈ E \ T . If e is a loop of G the latter I(M follows immediately. If e is no loop of G the argument follows from the fact that for every I ⊆ E \ {e} it holds I ∪ {e} contains a cycle of G if and only if I contains a cycle of G.e. 2

Proposition 3.55. Every minor of a matroid representable over a field F is again an F -representable matroid.

3.7. Submatroids

33

The following proof is an own result. ˆ (A|T ) = M ˆ (A)|T holds for every matrix A and Proof : By Remark 3.47, M any subset T of the index set of the columns of A. From the fact that the dual of an F-representable matroid is also F-representable and since contraction is the dual operation of restriction, it ˆ (A.T ) = M ˆ (A).T . follows that M 2 The last proof can be generalized to regular matroids. This yields the next proposition. Proposition 3.56. Every minor of a matroid representable over every field F is again representable over every field. By Proposition 3.55, a contraction of an F-representable matroid is Frepresentable. The question now is how a representation of such a contraction can be constructed. While restriction corresponds to removing columns, see Remark 3.47, it will turn out that contraction corresponds to removing certain rows and columns. ˆ (A) be a matrix matroid with representation maProposition 3.57. Let M trix A ∈ Mat(r × n, F). Let e ∈ {1, 2, . . . , n} and define T := {1, 2, . . . , n} \ ˆ (A).T obtained by con{e}. A representation matrix A.T for the matroid M ˆ (A) to T can be derived from A by performing the following: tracting M Delete the column index by e if it is a zero column. Transform the column indexed by e in a unit vector using elementary row operations. Such operations are called pivot operations. Next, delete this column together with the row containing the unique non-zero entry of it. Proof : Define A.e to be the matrix obtained from A by applying Proposition 3.57 to T := {1, 2, . . . , n} \ {e}. ˆ (A) then e is the index of a zero column and If e is a loop of M ˆ (A).e =M ˆ (A)|e. M ˆ (A) assume that e is the index of column 1 having a If e is no loop of M non-zero entry in row 1. This can be obtained by row and column swaps if necessary. Pivoting at this entry transforms A into a matrix where the

3.8. Excluded Minor Theorems

34

first column is equal to the first unit vector. Let I ⊆ {2, 3, . . . , n} with |I| = k ≤ n − 1. Then the set of columns indexed by I ∪ {e} is linearly independent if and only if the corresponding submatrix   1 a1 . . . ak 0     ..   . b1 . . . bk  0

where bi ∈ F r−1 , i ∈ I, has rank k + 1.
This is true if and only if the matrix b1 . . . bk has rank k. But this is true if and only if the columns of A.e indexed by I are linearly independent. ˆ (A).e) = I(M ˆ (A.e)) and therefore M ˆ (A).e =M ˆ (A.e). Hence, I(M 2

3.8

Excluded Minor Theorems

In this section, it is shown how minors can be used to define the different classes of matroids in an alternative way using the concept of excluded minors. Definition 3.58. Let M be a minor closed family of matroids. A matroid ˆ is called an excluded minor of M if M ˆ∈ M / M, but every proper minor of ˆ is in M. M The next lemma will be of interest for the characterization of matroids and follows from the fact that the dual of an F-representable matroid is also F-representable. ˆ is an excluded minor for F-representability Lemma 3.59. If the matroid M then Mˆ ∗ is also an excluded minor for F-representability. Note, that this only holds for F-representability and is not necessarily valid for subclasses of representable matroids such as graphic matroids. The following proposition and its proof is a collection of an idea presented in Kashyap [15].

3.8. Excluded Minor Theorems

35

Proposition 3.60. Let F be a collection of matroids. Define MF to be the ˆ such that no minor of M ˆ is in F. Then MF is minor set of all matroids M closed. ˆ is in a Proof : From Definitions 3.53 and 3.58, it follows that a matroid M ˆ is an excluded minor of minor closed family M if and only if no minor of M M. ˆ ∈ MF Take F as the set of all excluded minors of MF then no minor of M can be an excluded minor. 2 Robertson and Seymour [7] could have shown that any minor closed family of binary matroids must have finitely many excluded minors. This result enables to characterize classes of F-representable matroids. First, all excluded minors for the considered class of matroids have to be found. Then, one can say that all matroids that have no minor being isomorphic to these matroids are not contained in the considered class since by Proposition 3.55 every minor of an F-representable matroid is again F-representable. There are two criteria to identify an excluded minor. These criteria are the size of the field and the structure of the matroid. For F-representability, in general, the complete lists of excluded minors are not known. Nevertheless, for matroids that are representable over F2 or F3 the complete set of excluded minors can be listed. In the remainder of this section, some theorems that characterize classes of matroids in the described way, so called excluded minor theorems, are stated. For proofs the book of Oxley [18] is referred. In the first excluded minor theorem, the uniform matroids from Subsection 3.2.1 are important for the classification. ˆ is Fq -representable if it has no U2,q+2 or Uq,q+2 Theorem 3.61. A matroid M minor. Note that this theorem only states that the mentioned minors are excluded minors for a class of matroids but there might be other excluded

3.8. Excluded Minor Theorems

36

minors for the same class. Tutte [23] showed that U2,4 is the unique excluded minor for the class of binary matroids. Theorem 3.62 (Oxley, see [18], Chapter 6). A matroid is binary if and only if it has no U2,4 minor. In the next theorem, all excluded minors for the class of ternary matroids are enumerated. Theorem 3.63 (Oxley, see [18], Chapter 6). A matroid is ternary if and only if it has no minor isomorphic to any of the matroids U2,5 , U3,5 , F7 and F7∗ . F7 is called the Fano Matroid and ˆ (A) of the matrix M  1 0 0 A = 0 1 0 0 0 1

is defined to be the matrix matroid

 0 1 1 1 1 0 1 1 1 1 0 1

that has as columns all non zero vectors of F32 , see [24], Chapter1. The following excluded minor theorem characterizes the class of regular matroids. Theorem 3.64. A binary matroid is regular if and only if it has no minor isomorphic to the Fano Matroid or its dual. The last excluded minor theorem stated in this section gives a characterization of the classes of graphic cographic and planar matroids which are subclasses of the class of regular matroids. Theorem 3.65. (1) A regular matroid is graphic if and only if it has no ˆ (K5 ))∗ or (M ˆ (K3,3 ))∗ . minor isomorphic to (M (2) A regular matroid is cographic if and only if it has no minor isomorphic ˆ (K5 ) or M ˆ (K3,3 ). to M

3.8. Excluded Minor Theorems

37

(3) A regular matroid is planar if and only if it has no minor isomorphic ˆ (K5 ))∗ , (M ˆ (K3,3 ))∗ , M ˆ (K5 ) and M ˆ (K3,3 ). to any of the matroids (M ˆ (K3,3 ) is the cycle matroid of the complete bipartite graph on three M elements. The graph K3,3 is drawn in Figure 3.6 ([18], Some interesting matroids).

Figure 3.6: The complete bipartite graph on three elements

38

Chapter 4 Composition and Decomposition of Matroids This chapter deals with matroid sums that allow to decompose a bigger matroid into two smaller ones or vice versa. The stated definitions and results can be found in the book of Oxley [18] or in the articles of Seymour [20] or Gr¨otschel and Truemper [10]. For the case of Graphic Matroids detailed examples will be given. The chapter ends with important (De)Composition Theorems.

4.1

Sums of Matroids

In this section, different sums for matroids are introduced. Such an operation composes two smaller matroids to a bigger one. On the other side, a matroid can be decomposed in two smaller matroids if the first can be written as a sum of the two smaller matroids. The definitions of the sums as well as criteria for a matroid to be a sum of two other matroids are given in this section. The first sum that is introduced can be build if two matroids have distinct ground sets. ˆ 1 = (N1 , I1 ) and M ˆ 2 = (N2 , I2 ) be Theorem and Definition 4.1. Let M ˆ = (N, I) with N = N1 ∪ N2 and two matroids with N1 ∩ N2 = ∅. Then M I = {I1 ∪ I2 : I1 ∈ I1 , I2 ∈ I2 } is a matroid. This matroid is called the ˆ 1 and M ˆ 2 and is denoted by M ˆ 1 ⊕M ˆ 2. direct sum of M 39

4.1. Sums of Matroids

40

Moreover, it holds ˆ 1 ⊕M ˆ 2 ) = {B1 ∪ B2 : B1 ∈ B(M ˆ 1 ), B2 ∈ B(M ˆ 2 )}. (1) B(M ˆ 1 ⊕M ˆ 2 ) = C(M ˆ 1 ) ∪ C(M ˆ 2 ). (2) C(M ˆ 1 ⊕M ˆ 2 ) = C∗ (M ˆ 1 ) ∪ C∗ (M ˆ 2 ). (3) C∗ (M ˆ 1 ⊕M ˆ 2 )∗ =M ˆ ∗1 ⊕M ˆ ∗2 . (4) (M ˆ 1 ⊕M ˆ 2 is a matroid as well as the proofs of (1) to Proof : The fact that M (3) can be proofed straightforward. (4) follows from (2) and (3). 2 ˆ is the direct sum of M ˆ 1 and M ˆ 2 then, obviously, Observation 4.2. If M ˆ 1 and M ˆ 2 are proper minors of M ˆ. M A detailed example for a direct sum is given in Section 4.4. Before more sums can be defined the concept of Tutte k-connectedness must be introduced. ˆ = (N, I), for a positive integer k, a Definition 4.3. Given a matroid M ˆ if partition (T, T C ) of N is called a Tutte k-separation of M

and

 min |T | , T C ≥ k

r(T ) + r(T C ) − r(N ) ≤ k − 1

(4.1.1) (4.1.2)

The separation is called a minimal Tutte k-separation if equality holds in (4.1.1) and it is called an exact Tutte k-separation if equality holds in (4.1.2). ˆ is called Tutte k-connected if it has Definition 4.4. For k ≥ 2, a matroid M no Tutte l-separation for any l < k. Note that Tutte k-separations are defined for k ≥ 1 but Tutte kconnectedness is only defined for k ≥ 2.

4.1. Sums of Matroids

41

ˆ ) of a matroid M ˆ is defined as Definition 4.5. The Tutte connectivity λ(M ˆ ) := min{k : M ˆ is k-separated}. λ(M ˆ ) is set to ∞ if the required λ(M minimum does not exist. Example 4.6. Consider the cycle matroid of the complete graph K5 represented by the matrix A′ from Example 3.24. It can be seen that the partition (T1 , T1C ) of N = {1, 2, . . . , 10} with T1 = {1, 2, 4, 5, 7, 9} is an exact and minimal Tutte 4-separation but no Tutte 5-separation since T C = 4. Further, (T2 , T2C ) with T2 = {1, 2, 3, 4, 6, 8, 10} is an exact and minimal Tutte ˆ (K5 ). 3-separation of M ˆ (K5 ). It may be verified that there are no Tutte 1-,2- and 5-separations of M There exists no Tutte k-separation for k ≥ 6 since |N | = 10. ˆ (K5 ) has Tutte connectivity λ(M ˆ (K5 )) = 3 and By the above definitions M is Tutte 2- and 3- but not 4-connected. For Tutte k-connectedness, Tutte k-separability and Tutte connectivity the following observations can be made. ˆ is k-connected if and only if k ≤ λ(M ˆ ). Observation 4.7. (1) M (2) Any Tutte k-separation of a Tutte k-connected matroid has to be exact. (3) If a Tutte k-connected matroid has a Tutte k-separation it can not be Tutte k+1-connected, but it might have a Tutte k+1-separation. (4) If there is an exact Tutte k-separation of a matroid but no information about connectedness there might also exist a (exact) Tutte k+1- or a (exact) Tutte k-1-separation. (5) If there is no Tutte k-separation there is also no Tutte k-1-separation. (6) If there is a Tutte k-separation there exists an exact Tutte l-separation for some l ≤ k. Proposition 4.8. Tutte k-connectedness is invariant under the operation of taking duals.

4.1. Sums of Matroids

42

ˆ = (N, I). This is equivaProof : Let (T, T C ) be a Tutte k-separation of M lent to r(T ) + r(T C ) − r(N ) = k + 1. By the definition of the corank function this is equivalent to r(T ) + r∗ (T ) − |T | = k + 1. Again by the definition of the corank function this is equivalent to r∗ (T ) + r∗ (T C ) − r∗ (N ) = k + 1. But this means the same as (T, T C ) is a Tutte k-separation of Mˆ ∗ . 2 Definition 4.9. A Tutte 3-connected matroid is called internally Tutte 4connected if all its Tutte 3-separations are minimal. ˆ (K5 ), it turns out that all Tutte 3Considering more precisely M ˆ (K5 ) is Tutte 3-connected, it is also separations are minimal. Since M internally Tutte 4-connected. A Tutte 3-connected matroid has six or more elements. If such a matroid has a (co)circuit of three elements it can no longer be Tutte 4-connected but it can be internally Tutte 4-connected. The internally Tutte 4-connected matroids form a special class lying between the Tutte 3- and 4-connected matroids. Matroids can not only be classified by being representable over some field, but also by their Tutte k-connectedness as visualized in Figure 4.1. Next, connectedness is related to the direct sum. It will turn out that the direct sum is not only related to Tutte connectedness but also to another matroid connectedness which is defined next. ˆ = (N, I) is named connected if and only if Definition 4.10. A matroid M , for every pair of distinct elements of N , there exists a circuit containing both. By this definition, it is evident that a matroid is disconnected if and only if it is Tutte 1-separated. On the other side, a connected matroid is not Tutte 1-separated. Next, a relation to the direct sum is given. ˆ is Tutte 1-separated if and only if it is the direct Lemma 4.11. A matroid M sum of nonempty matroids.

4.1. Sums of Matroids

43

Figure 4.1: Classification of matroids according to Tutte connectedness ˆ. Proof : Let N be the ground set of the matroid M First, note that for any partition (T, T C ) of N it holds r(T ) + r(T C ) ≥ r(N ). ˆ if and only if T and From this, it follows that (T, T C ) is a 1-separation of M T C are not empty and r(T ) + r(T C ) = r(N ). The last is equivalent to C = C.T ⊕ C.T C . 2 From now on, matroid composition and decomposition is restricted to binary matroids. In the following, k-sums of matroids for k = 2, 3 are studied. First, the original definitions of 2- and 3-sums are stated together with the results of the investigations of Seymour [20] concerning both sums. The results of Seymour are very abstract and hold for general matroids. For binary matroids, Gr¨otschel and Truemper [10] developed a construction scheme for 2- and 3-sums. The construction scheme for 2-sums is presented in Section 4.2. Gr¨otschel and Truemper [10] developed not only a construction scheme for the 3-sum of Seymour but also a construction scheme for an additional 3-sum. To distinguish both sums, Gr¨otschel and Truemper call the 3-sum of Seymour ∆-sum and the additional 3-sum Y-sum. The

4.1. Sums of Matroids

44

construction schemes for 3-sum (de)compositions are presented in Section 4.3. First, a definition of the 2-sum introduced by Seymour is given. ˆ 1 and M ˆ 2 be two binary matroids on ground sets Definition 4.12. Let M N1 and N2 with min {|N1 | , |N2 |} ≥ 3. Assume that N1 ∩ N2 = {e} and {e} ˆ 1 or M ˆ 2 . Then the matroid M ˆ with ground set is no loop or coloop of M N = N1 ∆N2 with the property that all cycles are of the form C = C1 ∆C2 ˆ 1 and C2 is a cycle of M ˆ 2 is called a 2-sum of M ˆ1 where C1 is a cycle of M ˆ 2. and M The following theorem summarizes the investigations of Seymour with regard to 2-sums. A proof takes several pages in [20]. Theorem 4.13. If (T, T C ) is an exact Tutte 2-separation of a binary matroid ˆ then there exists a new element e, a binary matroid M ˆ 1 with ground set M ˆ 2 with ground set T C ∪ {e} such that M ˆ is a T ∪ {e} and a binary matroid M ˆ 1 and M ˆ 2 . Conversely, if M ˆ is a 2-sum of M ˆ 1 and M ˆ 2 with ground 2-sum of M ˆ. sets N1 and N2 then (N1 \ N2 , N2 \ N1 ) is an exact Tutte 2-separation of M ˆ 1 and M ˆ 2 are isomorphic to proper minors of M ˆ. Moreover, M Next, the definition of Seymour’s 3-sum is given. ˆ 1 and M ˆ 2 be two binary matroids on ground sets N1 Definition 4.14. Let M and N2 with N1 ∩ N2 = {e, f, g} and min {|N1 | , |N2 |} ≥ 7 such that {e, f, g} ˆ 1 and M ˆ 2 but contains no cocircuit of M ˆ 1 and M ˆ 2 . Then is a circuit of M ˆ with ground set N = N1 ∆N2 and where all cycles are of the the matroid M ˆ 1 and C2 is a cycle of M ˆ 2 is called form C = C1 ∆C2 where C1 is a cycle of M ˆ 1 and M ˆ 2. a 3-sum of M Analogous to the 2-sum, the next theorem is a summary of the investigation of Seymour concerning 3-sums. The proof of this theorem is very involved and can be found in [20]. Theorem 4.15. If (T, T C ) is an exact Tutte 3-separation of a binary matroid  ˆ with min |T | , T C ≥ 4 then there exists new elements e, f, g, a binary M ˆ 1 with ground set T ∪ {e, f, g} and a binary matroid M ˆ 2 with matroid M

4.1. Sums of Matroids

45

ˆ is a 3-sum of M ˆ 1 and M ˆ 2 . Conversely, ground set T C ∪ {e, f, g} such that M ˆ is a 3-sum of M ˆ 1 and M ˆ 2 and N1 and N2 are the ground sets of M ˆ1 if M ˆ 2 then (N1 \ N2 , N2 \ N1 ) is an exact Tutte 3-separation of M ˆ and and M ˆ is Tutte 3-connected, then min {|N1 \ N2 | , |N2 \ N1 |} ≥ 4. Moreover, if M ˆ 1 and M ˆ 2 are isomorphic to proper minors of M ˆ. M The above results of Seymour [20] are only statements about existence of an appropriate k-sum (de)composition. These statements give neither a ˆ or M ˆ1 description for the construction of the representation matrices of M ˆ 2 nor a hint how to find the new elements e or e, f, g respectively. This and M gap is filled by Gr¨otschel and Truemper in [10]. Among a 2-sum, Gr¨otschel and Truemper introduce two different sums, called ∆- and Y -sum, which will both turn out to be 3-sums. This implies that there are at least two different interpretations of a 3-sum having the properties investigated by Seymour and stated in Theorem 4.15. First results of the work of Gr¨otschel and Truemper [10] concerning exact Tutte k-separations are collected in the following proposition. Proposition 4.16. Let (T, T C ) be an exact Tutte k-separation of a binary ˆ for k ≥ 1. Then there exist a base X2 of the matroid M ˆ |T C and matroid M ˆ |T such that X := X1 ∪ X2 is a base an independent set X1 of the matroid M ˆ and the submatrix R of the standard representation matrix A = (I|R) of M ˆ corresponding to the base X is of the form of M

X1

Y1

Y2

A1

0 (4.1.3)

R= X2

D

A2

,where T = X1 ∪ Y1 , T C = X2 ∪ Y2 and the rank of the sub-submatrix D is equal to k−1. Conversely, any matrix R of the form 4.1.3 defines a k-separation (X1 ∪ Y1 , X2 ∪ Y2 ) if min {|X1 ∪ Y1 | , |X2 ∪ Y2 |} ≥ k. For this proposition, an own proof is proposed.

4.2. Construction of 2-Sum (De)Compositions

46

Proof : “⇒”: Let k ≥ 1 and let (T, T C ) be an exact k-separation of the ˆ represented by the binary matrix A. Then binary matroid M ˆ ) = k − 1. r(T ) + r(T C ) − r(M

(4.1.4)

ˆ |T C . Hence r(X2 ) = r(T C ). Choose an independent Let X2 be a base of M ˆ |T such that r(X1 ) = r(M ˆ ) − r(X2 ) = r(M ˆ ) − r(T C ) and set X1 of M ˆ . Such X1 exists since r(X2 ) ≤ r(M ˆ ) and X := X1 ∪ X2 is a base of M r(X2 ) = r(T C ). Define Y1 := T \ X1 and Y2 := T C \ X2 . Arrange the column indices of A as (X1 , X2 , Y1 , Y2 ). Since r(X2 ) = r(T C ) a transformation in standard form yields the matrix X1 ′

A = Ir(Mˆ )−r(T C ) 0

X2 0 Ir(T C )

Y1 A1 D

Y2 0 A2

(4.1.5)

By construction, the column submatrix indexed by Y1 has rank equal to ˆ the last is equal to r(T ) − r(T ) − r(X1 ). Since X = X1 ∪ X2 is a base of M ˆ ) − r(X2 )) which is the same as r(T ) − (r(M ˆ ) − r(T C )). But by (4.1.4) (r(M this is equal to k − 1. Since A′ is in standard form and r(T ) = r(X1 ) + r(Y1 ) the submatrix D must have rank k − 1. “⇐”: Let R be of the form (4.1.3) and let rank(D) = k − 1. Consider ˆ with ground set N = X1 ∪ X2 ∪ Y1 ∪ Y2 represented by the matroid M the matrix A = (I|R). Define T := X1 ∪ Y1 and T C := X2 ∪ Y2 . Assume  min T, T C ≥ k. Then r(T C ) = r(X2 ) + r(Y2 ) = r(X2 ) = |X2 | and r(T ) = r(X1 ∪ Y1 ) = r(X1 ) + k − 1 = |X1 | + k − 1. Together, it follows ˆ ) + k − 1 which implies that r(T ) + r(T C ) = |X1 | + k − 1 + |X2 | = r(M ˆ. (T, T C ) is an exact k-separation of M 2

4.2

Construction (De)Compositions

of

2-Sum

If a matrix matroid fulfills the conditions of Theorem 4.13 it has a representation matrix that has the same form as the matrix A from Proposition

4.2. Construction of 2-Sum (De)Compositions

47

ˆ 1 and M ˆ 2. 4.16. Thus, it can be written as a 2-sum of two binary matroids M ˆ 1 and M ˆ 2 as Gr¨otschel and Truemper [10] describe the construction of M follows. ˆ 1: Construction of M (1) Construct a new matrix R1e by taking the submatrix A1 of (4.1.3) and adding one non zero row from D which receives the new index e. This non zero row exists since the rank of D is equal to 1 by Proposition 4.16. Y1 R1e = X1

A1

e

(4.2.1)

a

(2) Extend R1e to a standard representation matrix of the form M1 = (I|R1e ). ˆ 1 is the first component of the 2-sum. The associated matroid M ˆ 2: Construction of M (1) Construct a new matrix R2e by taking the submatrix A2 of (4.1.3) and adding one non zero column from D which receives the new index e. This non zero column exists since the rank of D is equal to 1 by Proposition 4.16. The ith component of u is 1 if the ith row of D is equal to a and o else.

R2e =

X2

e

Y2

u

A2

(4.2.2)

(2) Extend R2e to a standard representation matrix of the form M2 = (I|R2e ). ˆ 2 is the second component of the 2-sum. The associated matroid M To refer to the element e according to which the two sum is constructed ˆ =M ˆ 1 ⊕e M ˆ 2 and call it e-sum. the two authors introduced the notation M

4.3. Construction of 3-Sum (De)Compositions

48

The complementary operation, the composition, is described as follows: 2-sum composition : (1) Calculate u · a = D. Compose A1 of R1e , A2 of R2e and D to a matrix R of the form (4.1.3). (2) Extend R to a standard representation matrix of the form A = (I|R). ˆ is the matroid obtained by M ˆ 1 ⊕e M ˆ 2. The associated matroid M ˆ 1 and M ˆ 2 and assuming that N1 and N2 are the correspondConsidering M ing ground sets it can be verified that N1 ∩ N2 = {e}. Furthermore, by the structure of R1e (4.2.1) and R2e (4.2.2) there can not exist an all zero column or a row containing a single one at position e in any representation matrices ˆ 1 and M ˆ 2 . Hence, e is no loop or coloop of M ˆ 1 or M ˆ 2 . It can also be verof M ˆ =M ˆ 1 ⊕e M ˆ 2 can be written as symmetric difference ified that any cycle of M ˆ 1 and a cycle of M ˆ 2 . Thus, assuming min {|N1 | , |N2 |} ≥ 3 the of a cycle of M considered e-sum is equal to the 2-sum introduced by Seymour.

4.3

Construction (De)Compositions

of

3-Sum

The construction for a 3-sum is more complicated. It also turns out, that such a construction is not unique. For a matroid fulfilling the assumptions of Theorem 4.15 there exists a base X ′ such that the corresponding standard representation matrix is of the form of A from Proposition 4.16. But, since there is the additional condition that  min |T | , T C ≥ 4 there exists a base X such that R is even of the form X1 R= X2

Y1

Y2

A1

0

a b

1 0

0 1

˜ D

u v

(4.3.1) A2

4.3. Construction of 3-Sum (De)Compositions The sub-submatrix D=

a b

1 0

0 1

˜ D

u v

49

(4.3.2)

has rank k − 1 = 2 by Proposition 4.16. The associated matroid can be decomposed into a 3-sum of two binary maˆ 1 and M ˆ 2. troids M Gr¨otschel and Truemper [10] describe the construction of two different 3ˆ∆ ˆ∆ sums. In the first 3-sum the components are denoted by M 1 and M 2 and ˆ Y1 and M ˆ Y2 . The in the second 3-sum the components are denoted by M corresponding 3-sums are denoted by ⊕∆ , called ∆-sum and ⊕Y , called Yˆ , i.e. sum. In both cases the composition yield the same matroid that is M ˆ =M ˆ∆ ˆ∆ ˆY ˆY M 1 ⊕∆ M 2 =M 1 ⊕Y M 2 .

4.3.1

Construction of ∆-Sum (De)Compositions

ˆ∆ Construction of M 1 : (1) Construct a new matrix R1∆ as follows. Take the submatrix A1 of (4.3.1), add the two rows of D containing a and b which receive the new indices e and f and add a new column indexed by g that contains two single one’s at the positions e and f .

R1∆ =

X1 e f

a b

Y1

g

A1

0

1 0

0 1

(4.3.3)

1 1

(2) Extend R1∆ to a standard representation matrix of the form M1∆ = ˆ∆ (I|R1∆ ). The associated matroid M 1 is the first component of the ∆sum. ˆ ∆: Construction of M 2

4.3. Construction of 3-Sum (De)Compositions

50

(1) Construct a new matrix R2∆ as follows. Take the submatrix A2 of (4.3.1), add the two columns of D containing u and v which receive the new indices e and f and add a new column indexed by g that is the sum of the columns indexed by e and f .

R2∆ = X2

e 1 0

f 0 1

g 1 1

u

v

w

Y2 A2

(4.3.4)

(2) Extend R2∆ to a standard representation matrix of the form M2∆ = ˆ∆ (I|R2∆ ). The associated matroid M 2 is the second component of the ∆-sum. ∆-sum composition : (1) Calculate u v ·

a ˜ =D b

˜ to a matrix R of the form and compose A1 of R1∆ , A2 of R2∆ and D (4.3.1). (2) Extend R to a standard representation matrix of the form A = (I|R). ˆ is the matroid obtained by M ˆ 1 ⊕∆ M ˆ 2. The associated matroid M ˆ 1 ⊕∆ M ˆ 2 is straightThe construction of a representation matrix A for M forward. The matrix operations can be interpreted as matroid operations as ˆ∆ follows. The set {e, f, g} symbolized by the ∆ forms a triangle in both M 1 ∆ ˆ and M 2 . During the composition these two triangles are identified first and removed after. Consequently, while the ∆-sum is performed first a matroid is created that contains the triangle {e, f, g}. Then this triangle is removed from this matroid by a sequence of contractions and restrictions such that the resulting matroid is represented by A. Therefore, the translation from matrix operations into matroid operations is not unique. A detailed example describing the processes of (de)composition for a graphical matroid will be

4.3. Construction of 3-Sum (De)Compositions

51

given in Section 4.6. If the described ∆-sum is compared with Seymour’s 3-sum definition it can be verified that both sums are the same. Indeed, assuming that N1 and ˆ ∆ and M ˆ ∆ and min {|N1 | , |N2 |} ≥ 7, it holds N2 are the ground sets of M 1 2 (1) N1 ∩ N2 = {e, f, g}. ˆ∆ (2) It is impossible to generate a representation matrix neither of M 1 nor ∆ ˆ 2 containing a row that is the incidence vector of a subset of of M {e, f, g} by the structure of R1∆ (4.3.3) and R2∆ (4.3.4). Therefore, ˆ∆ ˆ∆ {e, f, g} contains no cocircuit of M 1 and M 2 . ˆ∆ ˆ∆ (3) Any cycle C of M 1 ⊕∆ M 2 is of the form C = C1 ∆C2 , where Ci are ˆ∆ cycles of M i for i = 1, 2.

4.3.2

Construction of Y -Sum (De)Compositions

This subsection starts with the construction of the additional 3-sum given by Gr¨otschel and Truemper [10]. First, this sum is considered in detail and finally a definition of it is derived. ˆ Y1 : Construction of M (1) Construct a new matrix R1Y as follows. Take the submatrix A1 of (4.3.1), add the two rows of D containing a and b which receive the new indices r, s and add a new row indexed by t that is the sum of the rows indexed by r, s.

Y1 X1

A1

R1Y =

(4.3.5) r s t

a b c

1 0 1

0 1 1

4.3. Construction of 3-Sum (De)Compositions

52

(2) Extend R1Y to a standard representation matrix of the form M1Y = ˆ Y1 is the first component of the Y (I|R1Y ). The associated matroid M sum. ˆY : Construction of M 2 (1) Construct a new matrix R2Y as follows. Take the submatrix A2 of (4.3.1), add the two columns of D containing u and v which receive the new indices r, s and add a new row indexed by t that contains a one at the positions indexed by r, s, t and zero else.

t R2Y =

X2

r 1 1 0

s 1 0 1

u

v

Y2 0 A2

(4.3.6)

(2) Extend R2Y to a standard representation matrix of the form M2Y = ˆ Y2 is the second component of the (I|R2Y ). The associated matroid M Y -sum. Y -sum composition : (1) Calculate u v ·

a ˜ =D b

˜ to a matrix R of the form and compose A1 of R1Y , A2 of R2Y and D (4.3.1). (2) Extend R to a standard representation matrix of the form A = (I|R). ˆ is the matroid obtained by M ˆ 1 ⊕Y M ˆ 2. The associated matroid M ˆ 1 ⊕Y M ˆ 2 corresponds to that of the The representation matrix A for M ∆-sum. Here again, the translation from matrix operations into matroid

4.3. Construction of 3-Sum (De)Compositions

53

ˆ Y1 and M ˆ Y2 . operations is not unique. The set {r, s, t} forms a triad in both M While the Y -sum is performed these two triads have to be identified first and removed after. Why a Y is chosen to symbolize this triad will become clear in an example for a graphical matroid given in Section 4.6. If the Y -sum is considered in detail the following turns out. (1) N1 ∩ N2 = {r, s, t}. ˆ Y1 nor of (2) It is impossible to generate a representation matrix neither of M ˆ Y2 containing linearly dependent columns whose index set is a subset M of {r, s, t} by the structure of R1Y (4.3.5) and R2Y (4.3.6). Therefore, ˆ Y1 and M ˆ Y2 . {r, s, t} contains no circuit of M ˆ Y2 is of the form C = C1 ∆C2 , where Ci are ˆ Y1 ⊕Y M (3) Any cycle C of M ˆ Yi for i = 1, 2. cycles of M Hence the Y -sum can be defined in a similar way as the ∆-sum. ˆ 1 and M ˆ 2 be two binary matroids on ground sets N1 Definition 4.17. Let M and N2 with N1 ∩ N2 = {r, s, t} and min {|N1 | , |N2 |} ≥ 7 such that {r, s, t} ˆ 1 and M ˆ 2 but contains no circuit of M ˆ 1 and M ˆ 2 . Then is a cocircuit of M ˆ with ground set N = N1 ∆N2 and where all cycles are of the the matroid M ˆ 1 and C2 is a cycle of M ˆ 2 is called form C = C1 ∆C2 where C1 is a cycle of M ˆ 1 and M ˆ 2. a Y-sum of M Gr¨otschel and Truemper [10] state that Theorem 4.15 holds true if the 3-sum is replaced by the Y -sum and the triangle {e, f, g} is replaced by the triad {r, s, t}. In the following three sections, an interpretation of a (de)composition of graphic matroids via direct sum, 2- and 3-sum is given. This visualization will help to verify the different constructions introduced in the preceding sections.

4.4. Graphical Interpretation of Direct Sum (De)Composition 54

4.4

Graphical Interpretation of Direct Sum (De)Composition

Consider the graph from Figure 4.2. The vertex-edge incidence matrix A is given by (4.4.1).

Figure 4.2: Graphical interpretation of the direct sum of matroids (1)  1 0  A= 0 0 1

0 1 0 0 1

0 0 1 0 1

0 0 0 1 1

1 1 0 0 0

 0 0  1  1 0

(4.4.1)

ˆ is A standard representation matrix of the corresponding cycle matroid M given by (4.4.2). X1 1 0 ′ A = 0 1 0 0 0 0

X2 Y1 0 0 1 0 0 1 1 0 0 0 1 0

Y2 0 0 1 1

(4.4.2)

ˆ is 1-separated, indeed (T, T C ) with T = X1 ∪ Y1 and T C = X2 ∪ M ˆ 1 and M ˆ 2 such that Y2 is a 1-separation. Hence, there exist matroids M

4.4. Graphical Interpretation of Direct Sum (De)Composition 55 ˆ =M ˆ 1 ⊕M ˆ 2 by Lemma 4.11. It can easily be verified that M1 and M2 are M ˆ 1 and M ˆ 2 , see Theorem and Definition 4.1. It representation matrices for M ˆ 1 and M ˆ 2 are proper minors of M ˆ. is also evident that M X1 Y1 M1 = 1 0 1 0 1 1

X2 Y2 M2 = 1 0 1 0 1 1

(4.4.3)

ˆ 1 and M ˆ 2. Obviously, M1′ and M2′ are also representation matrices for M In addition, both matrices are vertex-edge incidence matrices of the graphs drawn in Figure 4.3. Thus, Figure 4.3 is a graphical interpretation of the ˆ. carried out direct sum decomposition of M X1 Y1 1 0 1 M1′ = 0 1 1 1 1 0

X2 Y2 1 0 1 M2′ = 0 1 1 1 1 0

Figure 4.3: Graphical interpretation of the direct sum of matroids (2) ˆ 1 and M ˆ 2 can be graphically Conversely, the direct sum composition of M interpreted by identifying the unique common vertex of the two graphs from Figure 4.3.

4.5. Graphical Interpretation of 2-Sum (De)Composition

4.5

Graphical Interpretation (De)Composition

of

56

2-Sum

For an example of a 2-sum decomposition, consider the graph from Figure 4.4 with vertex-edge incidence matrix A given by (4.5.1).

Figure 4.4: Graphical interpretation of a 2-sum of matroids (1) 

1 1 A= 0 0

0 1 1 0

0 0 1 1

 0 1  0 1

(4.5.1)

Adding the three first rows to the last row of A yields a zero row. This row can be deleted and the resulting matrix A′ still represents the same cycle ˆ by Remark 3.19. A standard representation matrix for M ˆ is given matroid M by A′′ .

  1 0 0 0 A′ = 1 1 0 1 0 1 1 0

X1 1 A′′ = 0 0

X2 Y1 0 0 0 1 0 1 0 1 1

If A′′ is tested for Tutte k-connectedness it can be verified that there is ˆ . Moreover, (T, T C ) with T = X1 ∪ Y1 = {1} ∪ {4} and no 1-separation of M

4.6. Graphical Interpretation of 3-Sum (De)Composition

57

ˆ. T C = X2 = {2, 3} forms an exact two separation of M ˆ 1 and M ˆ 2 are constructed acThe representation matrices M1 and M2 of M cording to the explanation given in Section 4.2. X1 M1 = 1 0

e Y1 0 0 1 1

X2 M2 = 1 0 0 1

e 1 1

(4.5.2)

X1 1 M1′ = 0 1

e Y1 0 0 1 1 1 1

X2 1 0 M2′ = 0 1 1 1

e 1 1 0

(4.5.3)

ˆ 1 and M ˆ 2 but they are, in addition, the M1′ and M2′ also represent M vertex-edge incidence matrices for the graphs drawn in Figure 4.5. Each of these two graphs contains the new element e in form of an edge indexed by e. The 2-sum composition can be interpreted by identifying the two common vertices of the two graphs from Figure 4.5 and by deleting the two edges indexed by e.

4.6

Graphical Interpretation (De)Composition

of

3-Sum

In general it is not trivial to decide whether or not a given matroid is Tutte k-connected. To give an example for a ∆- and Y -sum (de)composition a Tutte 3-connected graphic matroid is needed. Given such a matroid, every Tutte 3-separation is exact by Observation 4.7. According to the following Proposition, the cycle matroid of the graph shown in Figure 4.6 is Tutte 3-connected. Proposition 4.18 (Oxley, see [18], Chapter 8). Let G = (V, E) be a connected graph with |V | ≥ 3 and assume that G is not isomorphic to the

4.6. Graphical Interpretation of 3-Sum (De)Composition

Figure 4.5: Graphical interpretation of a 2-sum of matroids (2)

58

4.6. Graphical Interpretation of 3-Sum (De)Composition

59

complete graph K3 on three vertices. Then ˆ (G)) = min{κ(G), girth(G)}. λ(M κ(G) is the connectivity of the graph G and is defined to be equal to the number of elements of a minimal vertex cut of G. A vertex cut is a subset W ⊆ V of the vertex set of G such that the deletion of the vertices in W and all incident edges render G disconnected. The girth of a graph G is equal to the number of edges of a minimal cycle in G. For the graph in Figure 4.6 κ(G) = girth(G) = 3.

Figure 4.6: Graphical interpretation of a 3-sum decomposition of a matroid The matrix A is the vertex-edge incidence matrix of G 

    A=    

1 0 0 0 0 0 1

0 1 0 0 0 0 1

0 0 1 0 0 0 1

0 0 1 1 0 0 0

0 0 0 1 1 0 0

0 0 0 0 1 1 0

0 0 0 0 0 1 1

1 1 0 0 0 0 0

1 0 0 0 0 1 0

0 1 1 0 0 0 0

0 0 0 1 0 0 1

0 0 0 0 1 0 1

         

and the matrix A′ is a standard representation matrix of the correspondˆ. ing cycle matroid M

4.6. Graphical Interpretation of 3-Sum (De)Composition

X1 1 0 ′ A = 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

X2 0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

0 0 1 1 1 1

Y1 1 1 0 0 0 0

1 0 1 1 1 1

0 1 1 0 0 0

Y2 0 0 1 1 0 0

0 0 1 1 1 0

60

(4.6.1)

The reader may verify that (T, T C ) with T = X1 ∪ Y1 = {1} ∪ {7, 8, 9} and T C = X2 ∪Y2 = {2, 3, 4, 5, 6}∪{10, 11, 12} is an exact Tutte 3-separation ˆ. of M

4.6.1

Graphical Interpretation (De)Composition

of

∆-Sum

Performing the constructions explained in Section 4.3 yield the following ˆ 1 and M ˆ 2 of a matrices M1∆ and M2∆ that represent the two components M ∆-sum. ′ ′ The matrices M1∆ and M2∆ are the vertex-edge incidence matrices of corresponding graphs drawn in Figure 4.7.

M1∆

′ M1∆

X1 1 = 0 0

e f 0 0 1 0 0 1

X1 1 = 0 0 1

e f 0 0 1 0 0 1 1 1

Y1 g 0 1 1 0 0 1 0 1 1 0 1 1

0 0 1 1

Y1 1 1 0 0

1 0 1 0

g 0 1 1 0

M2∆

M2∆

1 0 = 0 0 0

′

1 0 = 0 0 0 1

0 1 0 0 0

0 1 0 0 0 1

X2 0 0 1 0 0 X2 0 0 1 0 0 0

0 0 0 1 0

0 0 0 1 0 0

0 0 0 0 1

0 0 0 0 1 0

e f 1 0 0 1 0 1 0 1 0 1 e f 1 0 0 1 0 1 0 1 0 1 1 1

g 1 1 1 1 1 g 1 1 1 1 1 0

1 1 0 0 0

1 1 0 0 0 0

Y2 0 1 1 0 0 Y2 0 1 1 0 0 1

0 1 1 1 0

0 1 1 1 0 1

4.6. Graphical Interpretation of 3-Sum (De)Composition

61

Figure 4.7: Graphical interpretation of a Delta-sum decomposition of a matroid Conversely, identifying the three common vertices and the edges indexed by e, f and g first and deleting these edges after is a graphical interpretation of a ∆-sum composition.

4.6.2

Graphical Interpretation (De)Composition

of

Y -Sum

The matrices M1Y and M2Y are obtained as described in Section 4.3.

X1 1 Y M1 = 0 0 0

r 0 1 0 0

s 0 0 1 0

t 0 0 0 1

0 0 1 1

Y1 1 1 0 1

1 0 1 1

t 1 0 Y M2 = 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

X2 0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

r 1 1 0 0 0 0

s 1 0 1 1 1 1

0 1 1 0 0 0

Y2 0 0 1 1 0 0

ˆ 1 and M ˆ 2 , indeed there is a row The set {r, s, t} forms a triad in both M containing a one at the positions r, s and t and zero else in both M1Y and M2Y .

0 0 1 1 1 0

4.6. Graphical Interpretation of 3-Sum (De)Composition

62

ˆ 1 and M ˆ 2 which are given by the vertex-edge The graphs corresponding to M ′ ′ incidence matrices M1Y and M2Y are drawn in Figure 4.8. The set {r, s, t} is the edge set of a cut in both graphs. The Y symbolizes these three edges together with the node Q.

M1Y

′

X1 1 0 = 0 1 0

r 0 1 0 0 1

s 0 0 1 0 1

t 0 0 0 1 1

0 0 1 1 0

Y1 1 1 0 0 0

1 0 1 0 0

M2Y

′

t 1 0 0 = 0 0 0 1

0 1 0 0 0 0 1

0 0 1 0 0 0 1

X2 0 0 1 1 0 0 0

0 0 0 1 1 0 0

0 0 0 0 1 1 0

r 1 1 0 0 0 0 0

s 1 0 0 0 0 1 0

0 1 1 0 0 0 0

Figure 4.8: Graphical interpretation of a Y-sum decomposition of a matroid The Y -sum composition can be interpreted as follows. First, identify the common vertex Q. Second, delete Q and identify the edges r, s and t of the former graphs to three new edges r′ , s′ and t′ which form a cut in the composed graph. Finally, contract r′ , s′ and t′ .

Y2 0 0 0 1 0 0 1

0 0 0 0 1 0 1

4.7. Properties of k-Sums

63

ˆ (K5 ) corresponding to the complete Remark 4.19. The cycle matroid M graph K5 is internally Tutte 4-connected. Consequently, it is not 1-separated ˆ (K5 ) can not be decomand has no exact Tutte 2- or 3-separation. Thus, M posed via direct sum, 2- or 3-sum For completeness, the discussion of 2- and 3-sums is terminated by some additional information, see [18], Chapter 7 and 12. Brylawski introduced the parallel- and the series connection of matroids in 1971 as well as the generalized parallel connection of matroids in 1975 before Seymour defined the 2- and 3-sum in 1980 . If the former two operations are compared with the 2-sum of Seymour it turns out that a restriction of a parallel connection or a contraction of a series connection coincides with the 2-sum composition. A restriction of a generalized parallel connection corresponds to Seymour’s 3-sum. In 1984, Robertson and Seymour [19] published a work on 2-sums and clique sums of graphs. The theory presented in this work is closed to the 2- and 3-sum of graphic matroids.

4.7

Properties of k-Sums

In this section, properties of the considered k-sums are studied. Used sources are the paper of Seymour [20] as well as the paper of Gr¨otschel and Truemper [10]. A first property of an e-sum decomposition is that any circuit or cocircuit ˆ can be constructed from circuits or cocircuits of M ˆ 1 and M ˆ 2 . Indeed, of M ˆ. by the definition of the e-sum, either (1) or (2) holds for a circuit C of M ˆ 1 or M ˆ 2. (1) C is a circuit of M ˆ 1 and a circuit C2 of M ˆ 2 each containing e (2) There exist a circuit C1 of M such that C = C1 ∆C2 . The considerations also hold if C, C1 and C2 are cocircuits.

4.7. Properties of k-Sums

64

An additional property stated in the next proposition is that the dual operation of the e-sum is again the e-sum. ˆ =M ˆ 1 ⊕e M ˆ 2 then Mˆ ∗ = M ˆ 1∗ ⊕e M ˆ 2∗ . Proposition 4.20. If M Since a proof was not contained in the literature an own one is proposed. ˆ =M ˆ 1 ⊕e M ˆ 2 with ground sets N, N1 , N2 , respectively. Then, Proof : Let M by definition, N = N1 ∆N2 and N1 ∩ N2 = {e}. Since {e} is no loop or ˆ 1 or M ˆ 2 it is also no coloop or loop of M ˆ 1∗ or M ˆ 2∗ . coloop of M ˆ to be the matroid on N = N1 ∆N2 with the property that every Define N ˆ is of the form D = D1 ∆D2 where Di are cocycles of M ˆ i for cycle D of N ˆ =M ˆ 1∗ ⊕e M ˆ 2∗ by definition of the e-sum and N ˆ =Mˆ ∗ by i = 1, 2. Then N ˆ. the construction of N 2 In the following two propositions it is shown that, as for a e-sum, any circuit or cocircuit of a ∆- or Y -sum can be constructed from circuits or cocircuits of its components. The propositions are a collection of the results of Seymour [20] as well as Gr¨otschel and Truemper [10]. ˆ =M ˆ 1 ⊕∆ M ˆ 2 and let {e, f, g} be the three common Proposition 4.21. Let M ˆ 1 and M ˆ 2 . If C is a circuit of elements of the ground sets N1 and N2 of M ˆ then one of the following statements holds. M ˆ 1 or M ˆ 2. (1) C is a circuit of M (2) There exists a unique z ∈ {e, f, g} such that Ci := (C ∩ (Ni \ {e, f, g})) ∪ ˆ i for i = 1, 2. Moreover, C = C1 ∆C2 . {z} is a circuit of M If C is a cocircuit instead of a circuit (1) remains valid and in (2) the element z has to be replaced by a unique two element subset Z ⊂ {e, f, g}. ˆ =M ˆ 1 ⊕Y M ˆ 2 and let {r, s, t} be the three common Proposition 4.22. Let M ˆ 1 and M ˆ 2 . If C is a circuit of elements of the ground sets N1 and N2 of M ˆ then one of the following statements holds. M ˆ 1 or M ˆ 2. (1) C is a circuit of M

4.7. Properties of k-Sums

65

(2) There exists a unique two element subset Ω ⊂ {r, s, t} such that Ci := ˆ i for i = 1, 2. Moreover, (C ∩ (Ni \ {r, s, t})) ∪ Ω is a circuit of M C = C1 ∆C2 . If C is a cocircuit instead of a circuit (1) remains valid and in (2) the two element set Ω has to be replaced by a unique element ω ∈ {e, f, g}. Two alternative proofs for the circuit case of Proposition 4.21 can be found in Seymour [20] and in the work of Gr¨otschel and Truemper [10]. The proof of Seymour is rather technical while the proof of Gr¨otschel and Truemper is based on matroid decomposition. This proof is only a sketched version which states the main arguments. Next, a detailed proof that uses the arguments of Gr¨otschel and Truemper is given for the circuit case of Proposition 4.22. The proof is presented for Proposition 4.22 since this proposition will be needed in Chapter 5. Proof for the circuit case of Proposition 4.22 : By the definition of the Y -sum, (1) holds if C contains only elements of N1 or N2 . ˆ is Now, assume that C contains one or more elements of N1 and N2 . Since M a Y -sum it has a standard representation matrix A = (I|R) where R is of the form (4.3.1). Define E1 := C ∩ (N1 \ {r, s, t}) and E2 := C ∩ (N2 \ {r, s, t}). Consider the submatrix

D = X2

Y1 a 1 b 0 ˜ D

0 1

u v

of (4.3.1). Define d to be the sum of all columns of D indexed by E1 ∩ Y1 . First, suppose that d = 0. Denote the ith column vector of A by pi for i ∈ N ˆ . Since C is a circuit of M ˆ the where N should denote the ground set of M |N | column vectors indexed by C are minimal dependent in F2 and it holds   0 X X X X  ..  X pi = pi + pi + pi + pi (4.7.1) . = i∈C i∈C∩X1 i∈C∩X2 i∈C∩Y1 i∈C∩Y2 0

4.7. Properties of k-Sums

66

By the definition of d  AC∩Y1  0  X   pi =  ..   .  

i∈C∩Y1

0

where AC∩Y1 is the sum of the columns of A1 indexed by C ∩Y1 and (0, . . . , 0)T is a zero vector of length |X2 |. By the structure of (4.3.1), it follows from (4.7.1) that   AC∩Y1  0  X X X   pi =  ..  and pi = pi  .  i∈C∩X1 i∈C∩X2 i∈C∩Y2 0

Hence, the column vectors of E1 = C ∩ (X1 ∪ Y1 ) and E2 = C ∩ (X2 ∪ Y2 ) |N | ˆ. are dependent in F2 but thisis a  contradiction to C being a circuit of M    1 0 1 Thus, d is equal to  0 , 1 or  1 , where w = u + v since the rank of w v u D equals 2.   1 ˆ 1 has a standard representation matrix Suppose d =  0 . The matroid M u M1 = (I|R1 ) where R1 is of the form (4.3.5). Denote the jth column vector of M1 by qj for j ∈ N1 . Then, by the choice of d   AC∩Y1 X  1   qj =   0  j∈C∩Y1 1

and since (4.7.1) still holds it follows by the structure of (4.3.5) that   AC∩Y1 X X  1   qj + qj =   0  j∈C∩X1 j∈C∩{r,s,t} 1

From this it follows again by (4.3.5) that C ∩ {r, s, t} = {r, t} and no other ˆ 1. choice is possible. Define Ω := {r, t}. Then C1 := E1 ∪ Ω is a circuit of M

4.7. Properties of k-Sums

67

ˆ 2 has a standard representation matrix M2 = (I|R2 ) where The matroid M R2 is of the form (4.3.6). Denote the kth column vector of M2 by lk for k ∈ N2 . From (4.7.1) and the structure of M2 it follows by the choice of d   0 X 1  lk =  0 k∈C∩(X2 ∪Y2 ) u Again by (4.7.1) and the structure of M2 it follows   0 X 1  lk =  0 k∈C∩{r,s,t} u

Hence, C ∩ {r, s, t} must be equal to {r, t} =: Ω and no other choice for Ω is ˆ 2. possible. Thus, C2:=E2 ∪ {e} isa circuit of M  1 0 The cases of d = 1 and d =  1  can be handled in the same manner. w v In the first case Ω = {s, t} and in the second case Ω = {r, s}. Obviously, C = C1 ∆C2 . 2 The cocircuit cases can be shown analogously using duality. Proposition 4.20 can not be extended to ∆- and Y -sums. Indeed, if one ˆ 1 and M ˆ 2 the other sum is not of both sums is defined for the matroids M ˆ 1 and M ˆ 2 changes defined. This results from the fact that dualization of M any circuit to a cocircuit and vice versa. However, it turns out that ⊕∆ and ⊕Y are dual operations as stated in the next proposition. ˆ= M ˆ 1∆ ⊕∆ M ˆ 2∆ then Mˆ ∗ = (M ˆ 1∆ )∗ ⊕Y (M ˆ 2∆ )∗ . Proposition 4.23. (1) If M ˆ= M ˆ 1Y ⊕Y M ˆ 2Y then Mˆ ∗ = (M ˆ 1Y )∗ ⊕∆ (M ˆ 2Y )∗ (2) If M A proof for this proposition was not given in the literature. Therefore, an own one is proposed. Proof :

4.7. Properties of k-Sums

68

(1) By Corollary 3.22, (4.3.3) and (4.3.4) are standard representation maˆ 1∆ )∗ and (M ˆ 2∆ )∗ are given by (4.7.2) and (4.7.3). trices for (M

(M1∆ )∗ =

X1

e

f

AT1

aT 1 0 1

bT 0 1 1

0

X2 1 0 uT 0 1 vT (M2∆ )∗ = 1 1 wT

Y1

e f

g (4.7.2)

I

g

I

Y2 (4.7.3)

AT2 Note that the number of columns of the identity matrix I in (4.7.2) is equal to |Y1 ∪ {g}| and the number of columns of the identity matrix I in (4.7.3) is equal to |{e, f, g} ∪ Y2 |. The triangle {e, f, g} becomes a triad under dualization. Indeed, in (4.7.2) the last row corresponds to the incidence vector of the set {e, f, g}. In (4.7.3) addition of the first three rows yields again the incidence vector of {e, f, g}. ˆ 1∆ )∗ and (M ˆ 2∆ )∗ fulfill both the conditions for a Y -sum, see Defini(M tion 4.17 ˆ ∆ )∗ and (M ˆ ∆ )∗ the following matroid opTo realize the Y -sum of (M 1 2 erations have to be made. First, the two common triads have to be identified to a single triad. Secondly, this triad has to be deleted. ˆ 1 ⊕Y M ˆ 2 , see [10], these matroid operSimilar to the construction of M ations can be translated into matrix operations by the following. Calculate aT

bT ·

uT ˜T =D vT

4.8. (De)Composition Theorems

69

˜ T to a matrix M ˜ of the and compose AT1 of (4.7.2) , A2 of (4.7.3) and D form (4.7.4). X1 ˜ = M

AT1

X2 aT 1 0

0

bT 0 1

Y1 ˜T D uT vT

Y2

I

(4.7.4)

AT2

˜ is also a standard representation maBy Corollary 3.22 and (4.3.1) M trix of Mˆ ∗ . (2) can be proved analogously. 2

4.8

(De)Composition Theorems

The following two theorems help to decide whether or not a k-sum (de)composition is possible. Both theorems follow from Theorem 4.13 or Theorem 4.15 together with Lemma 4.11. Theorem 4.24 (Oxley, see [18], Chapter 8). Every (binary) matroid that is not Tutte 3-connected can be constructed from Tutte 3-connected proper minors of it by a sequence of the operations of direct sum and 2-sum. The next theorem is an extension of the first containing the ∆- and Y sum, additionally. Theorem 4.25. Every binary matroid that is not internally Tutte 4connected can be constructed from Tutte 3-connected, internally Tutte 4connected proper minors of it by a sequence of the operations of direct sum, 2-sum, ∆-sum and Y -sum.

4.8. (De)Composition Theorems

70

These two theorems show the importance of a classification of matroids according to Tutte k-connectedness. The last theorem of this chapter is a very deep result of Seymour and the main result of [20]. The theorem is completed by the addition of Y sums which is possible since Gr¨otschel and Truemper showed that Seymour’s arguments remain true for the Y -sum. The theorem emphasizes the special role of regular matroids. ˆ can be constructed by means of Theorem 4.26. Every regular matroid M direct sums, 2-sums, ∆-sums and Y -sums starting with matroids each of ˆ , and each of which is either graphic, which is isomorphic to a minor of M cographic or isomorphic to R10 . This theorem has the following corollary. ˆ is a Tutte 3Corollary 4.27 (Oxley, see [18], Chapter 13). If M ˆ is graphic, connected, internally Tutte 4-connected, regular matroid, then M cographic, or isomorphic to R10 . R10 is a 10 element regular matroid that is neither graphic nor cographic having the following matrix as a standard representation matrix, see [20].

AR10



  =  

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 1 0 0 1

1 1 1 0 0

0 1 1 1 0

0 0 1 1 1

1 0 0 1 1

     

Theorem 4.26 is an alternative way to that of Theorem 3.64 to characterize the class of regular matroids. Note that the decompositions have not to be unique.

Chapter 5 Cycle Polytope of Binary Matroids In the first section of this chapter, an introduction to the theory of polyhedrons is given. In the second section, the cycle polytope of a binary matroid is defined. After that, this polytope is studied with regard to symmetry and dimension. The section ends with a complete description of the cycle polytopes of a certain subclass of binary matroids. In the third section, one more class of binary matroids whose corresponding cycle polytopes are called master polytopes is introduced. The master polytopes are considered under the same aspects as the polytopes in the former section. At the end of the third section, matroids that are obtained by contraction from matroids that corresponds to master polytopes are considered. In the following section, it is shown that facets of master polytopes can be lifted to facets of the cycle polytopes of certain other matroids. The fifth section is about the cycle polytope of binary uniform matroids. In the last section, the cycle problem of binary matroids is defined and it is investigated under which conditions this problem can be solved in polynomial time.

5.1

Polyhedral Theory

The aim of this section is to give an introduction to the theory of polyhedrons. Reference is made to the lecture notes of S. O. Krumke [16].

71

5.1. Polyhedral Theory

72

Roughly speaking, Polyhedral Theory is about subsets of Rn . To describe these subsets one needs not only the concept of linearly independence of vectors in Rn but also the concept of a particular independence which is defined in the following. Definition 5.1. Let ν1 , . . . , νk be vectors in Rn . A linear combination of P P these vectors of the form x = ki=1 λi νi such that ki=1 λi = 1 is called an affine combination. P The vectors ν1 , . . . , νk are called affinely independent if ki=1 λi νi = 0 and Pk i=1 λi = 0 implies that λ1 = λ2 = . . . = λk = 0.

The following lemma relates affinely independence and linearly independence.

Lemma 5.2. Let ν1 , . . . , νk be vectors in Rn . Then the following are equivalent (1) ν1 , . . . , νk are affinely independent. (2) ν2 − ν1 , . . . , νk − ν1 are linearly independent.     ν ν1 , . . . , k ∈ Rn+1 are linearly independent. (3) 1 1 Proof : Pk Pk (1)⇒(2) Assume i=2 λi (νi − ν1 ) = 0 and set λ1 := − i=2 λi . Then Pk Pk i=1 λi νi = 0 and i=1 λi = 0. By the affinely independence of ν1 , . . . , νk it follows λ1 = λ2 = . . . = λk = 0. Pk Pk (2)⇒(1) Assume that i=1 λi νi = 0 and i=1 λi = 0. Setting λ1 := Pk Pk − i=2 λi yields i=2 λi (νi − ν1 ) = 0. By the linearly independence of ν2 − ν1 , . . . , νk − ν1 it holds λ2 = . . . = λk = 0 but this implies λ1 = 0. (3)⇔(2) follows from the fact that k X i=1

 Pk    νi λ i νi = 0 i=1 Pk =0 ⇔ λi 1 i=1 λi = 0

5.1. Polyhedral Theory

73

2 Next, a particular subset of Rn is defined which is obtained by certain combinations of the elements of a smaller subset of Rn . Definition 5.3. Let X be a subset of Rn . The convex hull of X is defined to be the set of all convex combinations of vectors from X and is denoted by conv(X), i.e. conv(X) := {x =

k X i=1

λi νi : λi ≥ 0,

k X

λi = 1, ν1 , . . . , νk ∈ X}.

i=1

In the following definition, the main objects are introduced. Definition 5.4. A subset P of Rn that is described by finitely many inequalities is called a polyhedron. More formally P (A, b) := {x ∈ Rn : Ax ≤ b}, where A ∈ Mat(m × n, R) and b ∈ Rm . If a polyhedron is bounded it is called a polytope. Next, a technical tool that is important for the description of polyhedra is defined. Definition 5.5. Let P = P (A, b) be a polyhedron and let M be the index set of the rows of A. For a set S ⊆ P eq(S) := {i ∈ M : Ai x = bi ∀ x ∈ S}, where Ai denotes the i-th row of A is called the equality set of S. In the remainder of this chapter M always denotes the index set of the rows of A and Ai always denotes the i-th row of A. For a subset J ⊆ M AJ denotes all rows of A with indices in J. In what follows the question is how a system of inequalities that describes a polyhedron can be found. Given a polyhedron and a hyperplane in Rn , a first question is whether or not the polyhedron is located below this hyperplane.

5.1. Polyhedral Theory

74

Definition 5.6. Let S ⊆ Rn , w ∈ Rn and t ∈ R. The inequality wT x ≤ t is valid for S if S ⊆ {x : wT x ≤ t}. Next, certain subsets of a polyhedron are defined. These subsets contain only points belonging to the surface of the polyhedron. Definition 5.7. Let P ⊆ Rn be a polyhedron. The set F ⊆ P is called a face of P if there exists a valid inequality wT x ≤ t for P such that F = {x ∈ P : wT x = t}. F is called a proper face of P if F 6= ∅ and F 6= P . The following observations can be made for faces of polyhedra. Observation 5.8. (1) Since F = {x ∈ P : wT x = t} = {x : Ax ≤ b, wT x ≤ t, −wT x ≤ −t} every face of a polyhedron is again a polyhedron. (2) Any polyhedron P ⊆ Rn is a face of itself since oT x ≤ o is valid for any polyhedron and P = {x ∈ P : oT x = 0} is a face. (3) Since ∅ = P ∩ {x ∈ Rn : oT x = 1} ∅ is a face of any polyhedron P ⊆ Rn . The following theorem shows how face defining inequalities can be found. Theorem 5.9. Let ∅ 6= P = P (A, b) ⊆ Rn be a polyhedron. Then the set ∅ 6= F ⊆ Rn is a face of P if and only if F = {x ∈ P : AI x = bI } for a subset I ⊆ M. Proof : For the first part of the proof, it has to be shown that F is a face P of P for a subset I ⊆ M. The inequality cT x ≤ d with cT := i∈I AI and P d := i∈I bi is valid for P since AI x ≤ bI for all x ∈ P . Moreover, there exists at least one i ∈ I such that Ai x < bi for all x ∈ P \ F . Hence, F is a face of P . The second part of the proof is referred to [16]. 2 As a corollary, the following can be stated.

5.1. Polyhedral Theory

75

Corollary 5.10. Any polyhedron has finitely many faces. Proof : The statement follows from the fact that M is finite and Theorem 5.9. 2 Next, the dimension of a polyhedron is defined. Definition 5.11. The dimension dim P of a polyhedron P ∈ Rn is defined to be one less than the maximum number of affinely independent vectors in P . If dim P = n P is said to be full-dimensional. If P = ∅ then the dimension of P is defined to be −1. By the following theorem it is possible to determine the dimension of a face of a polyhedron. Theorem 5.12. Let P = P (A, b) ⊆ Rn be a polyhedron and let F 6= ∅ be a face of P . Then dim F = n − rank(Aeq(F ) ). Proof : From Linear Algebra, it is known that dim Rn = n = rank(Aeq(F ) ) + dim kern(Aeq(F ) ). Thus, it has to be shown that κ := dim kern(Aeq(F ) ) = dim F =: ϕ. κ ≥ ϕ : Let ν1 , ν2 , . . . , νϕ+1 be ϕ + 1 affinely independent vectors in F . Then by Lemma 5.2 the vectors νi − ν1 are linearly independent for i = 2, 3, . . . , ϕ + 1. Hence, Aeq(P ) (νi − ν1 ) = beq(P ) − beq(P ) = 0 for i = 2, 3, . . . , ϕ + 1. From this it follows ϕ ≤ κ. κ ≤ ϕ : ϕ ≥ 0 since F 6= ∅. Assume κ ≥ 0. It can be proved that there exists a point x˜ of F satisfying eq({˜ x}) = eq(F ), see [16]. Thus, Aeq(F ) x˜ = beq(F ) and AJ x˜ < bJ for J := M \ eq(F ). The last implies that there exists a basis {k1 ; . . . , kκ } of kern(Aeq(F ) ) and ǫ > 0 such that AJ (˜ x + ǫkl ) < bJ and Aeq(F ) (˜ x + ǫkl ) = beq(F ) for l = 1, . . . , κ. Hence, x˜ + ǫkl ∈ F for l = 1, . . . , κ. The vectors in {ǫk1 ; . . . , ǫkκ } are linearly independent by Lemma 5.2. Hence, {˜ x, ǫk1 + x˜, . . . , ǫkκ + x˜} is a set of affinely independent vectors in F . From this it follows ϕ ≥ κ. 2 Theorem 5.12 has the following corollary.

5.1. Polyhedral Theory

76

Corollary 5.13. Let ∅ 6= P = P (A, b) ⊆ Rn be a polyhedron. Then the following holds. (1) dim P = n − rank(Aeq(P ) ). (2) eq(P ) = ∅ implies dim P = n. (3) For any proper face F of P dim F ≤ dim P − 1. Proof : (1) follows from Theorem 5.12 together with Observation 5.8 (2). (2) follows from (1). (3) It is sufficient to show that rank(Aeq(F ) ) > rank(Aeq(p) ) for any proper face F of P since this implies dim F = n − rank(Aeq(F ) ) < n − rank(Aeq(P ) ) ≤ (dim P ) − 1 by Theorem 5.12. To prove this one can show that for j ∈ eq(F ) \ eq(P ) it holds Aj ∈ / span{Aeq(P ) } since from that it follows rank(Aeq(F ) ) ≥ rank(Aeq(P )∪{j} ) > rank(Aeq(P ) ). P Assume that Aj ∈ span{Aeq(P ) }, i.e. Aj = i∈eq(P ) λi Ai . Then for any xF ∈ F it holds b j = Aj x F =

X

λi Ai xF =

i∈eq(P )

X

λi bi .

(5.1.1)

i∈eq(p)

Since j ∈ / eq(P ) there exists x ∈ P such that Aj x < bj . But this yields a contradiction since, by (5.1.1), bj > Aj x =

X

i∈eq(P )

λi Ai x =

X

λi bi = bj .

i∈eq(P )

2 Next, a term for the proper faces of maximal dimension of a polyhedron is introduced. Definition 5.14. A proper face F of a polyhedron P = P (A, b) is called a facet of P if it is not strictly contained in any proper face of P .

5.1. Polyhedral Theory

77

The following theorem characterizes the facets of a polyhedron. Theorem 5.15. Let P = P (A, b) ⊆ Rn be a polyhedron and let F be a face of P . Then the following are equivalent (1) F is a facet of P (2) rank(Aeq(F ) ) = rank(Aeq(p) ) + 1 (3) dim F = dim P − 1 Proof : (2)⇔(3) follows from Theorem 5.12. (1)⇒(3) Assume that F is a facet of P and dim F < dim P − 1. Then rank(Aeq(P ) ) > rank(Aeq(P ) + 1 since (2)⇔(3). Choose i ∈ eq(F ) such that for J := eq(F ) \ {i} it holds rank(AJ ) = rank(Aeq(F ) ) − 1. Then G = {x ∈ P : AJ x ≤ bj } is a face of P containing F and dim G = dim F + 1 ≤ dim P − 1. This is a contradiction to F being a facet of P . (3)⇒(1) Suppose that G is a proper face of P strictly containing F . Then considering G as polyhedron and applying Corollary 5.13 to F yields dim F ≤ dim G − 1. Since dim F = dim P − 1 this implies that dim G = dim F . Again by Corollary 5.13, G can not be a proper face of P . 2 A corollary of Theorem 5.15 shows that any facet of a polyhedron P = P (A, b) is induced by at least one inequality in Ax ≤ b. Corollary 5.16. Let P = P (A, b) ⊆ Rn be a polyhedron and let F be a facet of P . Then F = {x ∈ P : Aj x = bj } for some j ∈ M \ eq(P ).

5.1. Polyhedral Theory

78

Proof : Choose j ∈ eq(F ) \ eq(P ) and define J := eq(p) ∪ {j}. Then the face G = {x ∈ P : AJ x ≤ bJ } induced by J can not be equal to P since for any point x not contained in any proper face of P it holds Aj x < bj . By the maximality of F , it must hold G = F . Thus F = {x ∈ P : AJ x = bJ } = {x ∈ P : Aeq(P ) x = beq(P ) , Aj x = bj } = {x ∈ P : Aj x = bj } 2 Given A and b all facets of P = P (A, b) can be described. The question is now, how many inequalities are needed for a complete description of a polyhedron. Definition 5.17. Let P = P (A, b) be a polyhedron. The system Ax ≤ b of equalities describing P is called irredundant if there exists no constraint Ai x ≤ bi such that P (A, b) = P (AM\{i} , bM\{i} ). If such a constraint exists it is called redundant. Note, that if there are more than one redundant constraints the removal of one redundant constraint might make the hole system irredundant. The following theorem shows that facets are necessary and sufficient to describe a polyhedron. Consequently a list of all facet defining inequalities is an irredundant system describing a polyhedron. Theorem 5.18. Let P = P (A, b) be a polyhedron. Define J := M \ eq(p) and assume that AJ x ≤ bJ is irredundant. Then for each row Aj of AJ the inequality Aj x ≤ bj defines a distinct facet of P . Conversely, for every facet F of P there exists a unique inequality in AJ x ≤ bJ inducing F . Proof : Let F be a facet of P . Then by Corollary 5.16 F = {x ∈ P : Aj x = bj } for some j ∈ J. Therefore, any facet is represented by an inequality in AJ x ≤ bJ . Again by Corollary 5.16 facets induced by distinct rows are also distinct. To prove that F = {x ∈ P : Aj x = bj } is a facet for j ∈ J it has to be

5.1. Polyhedral Theory

79

shown that dim F = dim P − 1. Since j ∈ eq(F ) \ eq(P ) it follows F 6= P . Hence, dim F ≤ dim P − 1. To show that dimF ≥ dim P −1 it has to be proved that eq(F ) = eq(P )∪{j} since then rank(Aeq(F ) ) ≤ rank(Aeq(P ) ) + 1. Take a point x˜ ∈ P that is not contained in any proper face of P . Then Aeq(P ) x˜ = beq(P ) and AJ x˜ < bJ . Define K := J \ {j}. Since Aj x ≤ bj is not redundant in Ax ≤ b there exists y such that Aeq(P ) y = beq(P ) , AK y ≤ bK and Aj y > bj . Let z be a convex combination of x˜ and y, i.e. z = λy + (1 − λ)˜ x for any λ ∈ (0, 1). Then Aeq(P ) z = beq(P ) , AK z < bK and Aj z = bj . Hence, z ∈ F and eq(F ) = eq(P ) ∪ {j}. 2 From Theorem 5.18 the following can be concluded for faces. Corollary 5.19. Any face F 6= P of a polyhedron P is the intersection of facets of P . Proof : Any face F of P is of the form F = {x ∈ P : AI x = bI } for an I ⊆ M, by Theorem 5.9. This is equivalent to F = {x ∈ P : AI\eq(P ) x = bI\eq(P ) } for the same I ⊆ M. And this is equivalent to T F = j∈I\eq(P ) {x ∈ P : Aj x = bj } again for the same I ⊆ M. Each of the sets in the intersection defines a facet, by Theorem 5.18. 2 In the following, it is shown that a polyhedron is not only characterized by its faces but also by its extreme points. Definition 5.20. An extreme point of a polyhedron P = P (A, b) ⊆ Rn is a point x∗ ∈ P that can not be written as a convex combination of two other points x, y of P , i.e. x∗ = λx + (1 − λ)y for some x, y ∈ P and λ ∈ (0, 1) implies that x = y = x∗ . By this definition it follows that an extreme point is a zero-dimensional face of P . Thus, by Corollary 5.10 any polyhedron has finitely many extreme points. The following theorem states how a polytope can be described in an alternative way. Theorem 5.21. Every polytope is the convex hull of its extreme points.

5.1. Polyhedral Theory

80

For a proof [16] is referred. Conversely, the convex hull of a finite set is always a polytope. Both results are summarized in the following theorem. Theorem 5.22. A polyhedron P ⊆ Rn is a polytope if and only if P = conv(X) for X ⊆ Rn finite. Proof : “⇒”: follows from Theorem 5.21 and the fact that any polytope has finitely many extreme points. “⇐”: Let X = {x1 , . . . , xl } ⊆ Rn be a finite set and P = conv(X). The proof consists of to steps. In step 1 an appropriate polytope P˜ is constructed such that in step 2 it can be shown that P˜ = P . Consider the bounded subset Q ⊆ Rn+1 which is defined as follows    a n T : a ∈ [−1, 1] , t ∈ [−1, 1], a x ≤ t ∀ x ∈ X . Q := t     a1 a Obviously, Q is a polytope. Let EP := ,..., p be the extreme t1 tp points of Q. Then by Theorem 5.21 Q = conv(EP ). Define P˜ := {x ∈ Rn : aTj x ≤ tj , j = 1, . . . , p}. “P ⊆ P˜ ” : Let x˜ ∈ P then x˜ can be written as convex i.e.   combination, Pk Pk aj ∈ Q and fix j x˜ = i=1 λi xi for x1 , . . . , xk ∈ P and i=1 λi = 1. For tj P P it holds aTj xi ≤ tj ∀i. Hence aTj x˜ = ki=1 λi aTj xi ≤ ki=1 λi tj = tj . Thus, aTj x˜ ≤ tj ∀j and x˜ ∈ P˜ . “P ⊆ P˜ ” : Assume that there exists y ∈ P˜ \ P . Then there exists a hyperplane wT x = t such that wT x ≤ t ∀x ∈ P and wT y > t, see e.g.   w˜ [16]. Scaling w, t by an appropriate ǫ > 0 yields ˜ ∈ Q such that t   w˜ w˜ T x ≤ t˜ ∀x ∈ P and w˜ T y > t˜. Since ˜ ∈ Q it can be written as convex t     Pp w˜ a combination of the extreme points of Q, i.e. = j=1 λj j . Since tj t˜

5.2. Cycle Polytopes y ∈ P˜ it holds aTj y ≤ tj ∀j. Thus w ˜T y = is a contradiction.

5.2

81 Pp

j=1

λj aTj y ≤

Pp

j=1

λj tj = t which 2

Cycle Polytopes

Cycle polytopes of binary matroids were studied by Barahona and Gr¨otschel [2] as well as Gr¨otschel and Truemper [10, 11]. In this section, the main results of the cited works are summarized. The cycle polytope of a binary matroid is defined as follows. ˆ be a binary matroid. Assume that N with |N | = n Definition 5.23. Let M ˆ . Then the cycle polytope P (M ˆ ) of the matroid M ˆ is is the ground set of M ˆ , i.e. defined to be the convex hull of the incidence vectors of all cycles of M ˆ ) = conv{xC ∈ Rn : C is a cycle of M ˆ }. P (M ˆ ) follows from Theorem 5.22. The well-definedness of P (M ˆ ) is the convex hull of all elements In other words, the cycle polytope P (M ˆ ) of the binary matroid M ˆ. of the circuit space C(M ˆ ) is equal to the null space of a binary representaBy Proposition 3.38 C(M ˆ and therefore a subspace of Fn2 . Since this null space is tion matrix A of M ˆ ) is the convex hull of the kernel of the linear mapping defined by A, P (M all the binary solutions of Ax ≡ 0 mod 2. ˆ is graphic then P (M ˆ ) is the convex hull of the incidence vectors of If M all Eulerian subgraphs of the corresponding graph G. ˆ is cographic then P (M ˆ ) is the convex hull of the incidence vectors of If M ˆ ) is also called all edge cuts of a corresponding graph H. In this case P (M cut polytope. Note that, in general, H is not the graph corresponding to the ˆ. cycle matroid of M ˆ ) are considered a very useful theorem is Before faces and facets of P (M stated. This theorem enables to derive a new face defining inequality from a

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ˆ ). given face defining inequality by taking advantage of the symmetry of P (M The theorem also states that the new face has the same dimension than the former one. Consequently, the theorem is a useful tool to find more faces of a cycle polytope if at least one face is known. Theorem 5.24 (Barahona and Gr¨ otschel, see [2]). If aT x ≤ α defines ˆ ) of dimension d, and C is a cycle of M ˆ , then the inequality a face of P (M ˆ ) of dimension d, where a ˜T x ≤ α ˜ also defines a face of P (M ( ae if e ∈ /C a ˜e := −ae if e ∈ C and α ˜ := α − aT xC . Proof : ˆ) : a ˜T x ≤ α ˜ is valid for P (M ˆ such that a Suppose that B is a cycle of M ˜ T xB > α ˜ . Let C be an other cycle ˆ ˆ, of M . Then by Theorem 3.32 xB∆C is the incidence vector of a cycle of M ˆ ) and i.e. an extreme point of P (M aT xB∆C = aT xB\C + aT xC\B = aT xB\C + aT xC − aT xB∩C =a ˜ T x B + aT x C > α ˜ + aT xC = α which is a contradiction to the validity of aT x ≤ α. The dimension of the face defined by a ˜T x ≤ α ˜ is equal to d: Since the dimension of the face defined by aT x ≤ α is equal to d there exists ˆ such that the corresponding incidence vectors d+1 cycles D1 , . . . , Dd+1 of M xD1 , . . . , xDd+1 are affinely independent. Hence, aT xDi = α for i = 1, . . . , d+1, see Definition 5.11. For i = 1, . . . , d + 1 define Fi := Di ∆C. Then a ˜T xFi = a ˜T xDi \C + a ˜T xC\Di = aT xDi \C − aT xC\Di = aT xDi − aT xC∩Di − (aT xC − aT xDi ∩C ) = α − aT xC = α ˜ Suppose that all xFi are affinely dependent. Then it might be assumed that P Pd+1 xF1 = d+1 i=1 λi xF −i and i=1 λi = 1. Then, by Remark 3.35 xD1 = xF1 ∆C = xF1 + xC =

d+1 X i=1

λi (xFi + xC ) =

d+1 X i=1

λi xDi

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83

which is a contradiction to xD1 , . . . , xDd+1 being affinely independent. There might be more than d + 1 affinely independent vectors in the considered face. Therefore the dimension is at least d. But if one assumes that the new face has dimension greater than d one can analogously show that the dimension of the former face must also be greater than d. 2 The following corollary of Theorem 5.24 is a symmetry statement about ˆ ). the extreme points of P (M ˆ ) ,and let F (v, d) and Corollary 5.25. Let v, w be two extreme points of P (M F (w, d) be the sets of faces of dimension d that contain v and w respectively. Then the cardinalities of F (v, d) and F (w, d) are equal. ˆ ) both are the incidence Proof : Since v and w are extreme points of P (M ˆ , call them V and W . Let F = {x ∈ P : aT x = α} vectors of cycles of M ˆ ). Theorem 5.24 applied to the cycle be a d-dimensional face of P (M C := W ∆V yields that F˜ = {x ∈ P : a ˜T x = α ˜ } is again a face of ˆ ) of dimension d. From the proof of Theorem 5.24 it follows that P (M xV ∆C = xV ∆W ∆V = xW = w is contained in F˜ . Hence, there exists a one to one correspondence between the elements of F (v, d) and the elements of F (w, d). 2 As a consequence of Corollary 5.25 it is possible to derive all facets of ˆ ) if all facets of P (M ˆ ) that contain a given extreme point are known. P (M ˆ ) can be completely described if all Since ∅ is a cycle of every matroid P (M the facets containing 0 are known. These facets are called homogeneous. The question now is how facet defining inequalities can be found. Before an answer to this question is given a system of inequalities which are ˆ with ground set N is valid for the cycle polytope of any binary matroid M presented. The so called trivial inequalities 0 ≤ xe ≤ 1 ∀ e ∈ N

(5.2.1)

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84

ˆ ) since P (M ˆ ) is contained in the unit hypercube. are always valid for P (M ˆ ) is given by A second class of inequalities which are always valid for P (M the following proposition. Proposition 5.26. The inequalities x(F )−x(D \F ) ≤ |F |−1 for all cocircuits D ⊆ N and all F ⊆ D, |F | odd (5.2.2) P where x(F ) := e∈F xe are called cocircuit inequalities and are valid for the cycle polytope of any binary matroid. A proof for this proposition was not contained in the literature. Therefore, an own proof is proposed. ˆ be a binary matroid. Let C be a cycle and D be a cocycle Proof : Let M ˆ . Then the cardinality of C ∩ D is even by Theorem 3.32. Thus, there of M are two cases that have to be distinguished. First assume that F ⊂ D and F ∩ C = ∅. This implies |D| > |F | and thus x(F ) − x(D \ F ) = |F | − (|D| − |F |) = 2 |F | − |D| ≤ |F | − 1. Second assume that F ⊆ D and F ∩ C 6= ∅. This implies |F ∩ D| ≤ |F |. But since |C ∩ D| is even and |F | is odd |F | must be less than |D|. Hence, again, x(F ) − x(D \ F ) ≤ |F | − 1. 2 Consider the set ˆ ) := {x ∈ R|N | : x fulfills (5.2.1) and (5.2.2)} Q(M

(5.2.3)

which is a polytope by Definition 5.4. Since the inequalities (5.2.1) and ˆ (5.2.2) are valid for the cycle polytope of any binary matroid M ˆ ) ⊆ Q(M ˆ) P (M ˆ. for any binary matroid M ˆ )=Q(M ˆ ) if all inequalities in (5.2.1) and (5.2.2) By Corollary 5.25 P (M which are fulfilled with equality by the all zero vector x = 0 define all the ˆ ) containing x = 0. facets of P (M Seymour [21] called this condition the sum of circuits property and showed

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ˆ has this property if and only if it has no F7∗ , M ˆ (K5 )∗ that a binary matroid M or R10 minor. ˆ )=Q(M ˆ ) if and only if M ˆ has no F7∗ , M ˆ (K5 )∗ or R10 Theorem 5.27. P (M minor. For a proof [21] is referred. From the above considerations, it follows that the inequalities (5.2.1) and ˆ ) completely if M ˆ has the sum of circuits property. (5.2.2) describe P (M However, some of these inequalities might be redundant. Therefore, criteria that help to decide whether or not one of the inequalities (5.2.1) and (5.2.2) ˆ ) are developed in the following. First of all, it is of defines a facet of P (M ˆ ) since by Theorem 5.15 the dimension interest to know the dimension of P (M ˆ ). of a facet is one less than the dimension of P (M ˆ = (N, I) be a binary matroid and |N | = n. Theorem 5.28. Let M ˆ ) is given by The dimension of the cycle polytope P (M ˆ ) = n − |G| − dim P (M

S X

(|Hi | − 1)

(5.2.4)

i=e

where G is the set of all coloops and Hi for i = 1, . . . , S are the coparallel ˆ . Or, equivalently, dim P (M ˆ )= S, i.e. the dimension is equal classes of M ˆ. to the number of coparallel classes of M A definition of a coparallel class is given by the dualization of Definition 3.4. Before a proof of Theorem 5.28 is given an example is presented to make the statement more comprehensible.   1 1 0 0 0 1 1 0  Example 5.29. Let A =  1 0 1 0 be a representation matrix of a 0 0 0 1 binary matroid with ground set N = {1, 2, 3, 4} that is the collection of the indices of the columns of A. ˆ . Thus, all By Proposition 3.45 the rows of A span the cocircuit space of M cocircuits of two elements are {1, 2}, {2, 3}, {1, 3} and {4} is the single coloop

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86

ˆ . H1 = {1, 2, 3} is a maximal subset of N that contains no coloop of M and where any two distinct elements are coparallel. Hence, H1 the single ˆ . Thus, dim(P (M ˆ (A))) by Theorem 5.28. Indeed, x1 = coparallel class of M (0, . . . , 0)T and x2 = (1, 1, 1, 0)T are all binary solutions of Ax ≡ 0 mod 2. ˆ has no coloops and no coparallel elements all subsets of N containIf M ing a single element are coparallel classes. Proof of Theorem 5.28: From the definition of a coparallel class it follows ˆ . Hence n = that (G, H1 , . . . , HS ) is a partition of the ground set N of M PS PS |G| + i=1 |Hi | and n − |G| − i=1 (|Hi | − 1) = S. From the cocircuit inequalities (5.2.2) and the trivial inequalities (5.2.1) it follows that ˆ and xe = 0 if D = {e} is a coloop of M ˆ. xe = xf if e, f are coparallel in M ˆ) Hence, for any vector x ∈P (M ˆ and xe = 0 if e ∈ N is a coloop of M

(5.2.5)

xe1 − xei = 0 for i = 2, .., k if H = {e1 , e2 , . . . , ek } ˆ. is a coparallel class in M

(5.2.6)

Again by the definition of a coparallel class, the inequalities given by (5.2.5) ˆ ) is at and (5.2.6) are linearly independent. Thus the dimension of P (M ˆ )= S Barahona and Gr¨otschel show that for most S. To prove that dim P (M ˆ ) it follows an additional equation aT x = α which is satisfied by all x ∈P (M a = 0 and α = 0, see [2]. Hence, there exists S + 1 affinely independent ˆ ). vectors in P (M 2 ˆ ) is Remark 5.30. From the above theorem, it follows immediately that P (M ˆ has no coloops and no coparallel classes. In this case, full-dimensional if M ˆ ) defines a facet, every inequality of an irredundant system describing P (M by Theorem 5.18.

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87

ˆ )=Q(M ˆ ) then all inequalities given by (5.2.1) and (5.2.2) define If P (M ˆ ) completely. However, not all inequalities given by (5.2.1) and (5.2.2) P (M ˆ ), i.e. the inequality system (5.2.1), (5.2.2) has has to define facets of P (M not to be irredundant. ˆ ) can be obtained is described in [2]. How a minimal system describing P (M The results are summarized in the following theorem. ˆ with ground set N has no F7∗ , M ˆ (K5 )∗ or Theorem 5.31. The matroid M R10 minor if and only if the system xe = 0 for every coloop e ∈ N

(5.2.7)

xe1 − xei = 0 for every coparallel class H = {e1 , e2 , . . . , ek } and every i ∈ {2, 3, . . . , k} ¯ such that e 0 ≤ xe ≤ 1 for every e ∈ N ˆ¯ does not belong to a triad of M ˆ¯ with x(F ) − x(D \ F ) ≤ |F | − 1 for every cocircuit F of M

(5.2.8) (5.2.9)

no chord and every F ⊆ D, |F | odd (5.2.10) ˆ ). is an irredundant system describing P (M ˆ¯ is the matroid obtained from M ˆ by deleting all coloops and Note that M ¯ is contracting {2, 3, . . . , k} for each coparallel class H = {e1 , e2 , . . . , ek }. N ˆ¯ . A chord is defined as follows. the ground set of M Definition 5.32. Let C ⊆ I be a cycle (respectively a circuit) of the matroid ˆ . An element e ∈ N \ C is called a chord of C if there exist two cycles M (respectively circuits) C1 and C2 such that C1 ∩ C2 = {e} and C1 ∆C2 = C. For a proof of Theorem 5.31 [2] is referred. ˆ has the From Theorem 5.31 it follows that, under the assumption that M sum of circuits property as well as no coloops and no coparallel classes, each ˆ ) if and only if e does not belong inequality in (5.2.1) defines a facet of P (M ˆ and each inequality in (5.2.2) defines a facet of P (M ˆ ) if and to a triad of M only if the cocircuit D has no chord.

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88

Remark 5.33. If the assumptions of Theorem 5.31 are fulfilled there exists ˆ ), but the number of a complete and irredundant system describing P (M ˆ. inequalities may be exponential in the cardinality of the ground set of M Gr¨otschel and Truemper [10] showed that the class of the matroids with the sum of circuits property is closed under e-sum and Y -sum but not under ∆-sum. From this, they derived the following decomposition theorem for matroids with the sum of circuits property. ˆ that has no F7∗ , M ˆ (K5 )∗ or R10 miTheorem 5.34. Any binary matroid M nor can be constructed by a sequence of 2-sums and Y -sums from graphic ˆ (K3,3 )∗ or M ˆ (V8 )∗ . matroids and matroids isomorphic to F7 , M ˆ (V8 )∗ is the dual of the cycle matroid of the graph V8 drawn in Figure M 5.1.

Figure 5.1: The graph V8

5.3

Master Polytopes

In this section, a new class of binary matroids is introduced. As for the binary matroids with the sum of circuits property the cycle polytope of a

5.3. Master Polytopes

89

matroid from the new class can be completely described. The new cycle polytopes have the interesting property that their facet defining inequalities also describe facets of the cycle polytopes of certain other binary matroids. This section summarizes the results from Gr¨otschel and Truemper [11]. First, the new class of matroids and its cycle polytopes are defined. Definition 5.35. For k ≥ 2, a complete binary matroid of order k is defined   to be the matroid represented by the matrix I Ak , where I is an identity matrix of order 2k − k − 1 and Ak ∈ Mat(2k − k − 1 × k, F2 ) is the binary matrix with k columns that has as rows all possible distinct 0-1 vectors except for the k unit vectors and the zero vector. For k = 1 the complete binary matroid consists of just one loop. The complete binary matroids are denoted by Lk , for k ≥ 1. The cycle polytopes of the complete binary matroids are called master polytopes.   For example, A2 = 1 1 and L2 consists of just one triad.   1 1 0 1 1 1  A3 =  1 0 1 0 1 1

Thus, L3 is isomorphic to F7∗ which is the dual of the Fano Matroid F7 . For k ≥ 2, Ak+1 can be obtained from Ak be the following   k A 0 Ak+1 =  Ak 1  , I 1

where 1 stands for a column vector of appropriate length consisting of only 1’s.

Note that the order of the rows of the Ak ’s is not unique. Hence, there are (2k − k − 1)! isomorphic complete binary matroids for any k ≥ 3. The following can be observed for complete binary matroids.

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90

(1) Any Lk has no coloops and no coparallel classes. Hence, any master polytope is full-dimensional by Theorem 5.12. (2) Any Lk has corank k. (3) Complete binary matroids are maximal with the properties (1) and (2). (4) Any Lk can be obtained from some Lk+1 by contractions only. (Note that for a zero column contraction is equal to restriction.) A minor that is obtained by contractions only, is called a contraction minor. Some more properties of complete binary matroids are collected in the following proposition. Proposition 5.36. (1) For k ≥ 3, any circuit of Lk consists of at least one element from the index set of the columns of Ak and of at least one element from the index set of the columns of the identity matrix I. (2) In every column of Ak there are 2k−1 − k zero’s and 2k−1 − 1 one’s, for k ≥ 2. (3) Every circuit C of Lk has cardinality 2k−1 . (4) Every nonempty cycle of Lk is actually a circuit. (5) For k ≥ 2, any Lk has 2k cycles. For this proposition an own proof is proposed. Proof : (1) Obviously, C is a circuit of Lk if and only if the elements of C are the indices of the columns of an minimal Eulerian column submatrix of   I Ak , i.e. a minimal column submatrix where each row has an even number of one’s. By the definition of Ak , no column submatrix of Ak can be Eulerian. Hence, a circuit can not only consist of elements from the index set of the columns of Ak . Conversely, a circuit of Lk can not only consist of elements from the index set of the columns of I since in I all columns are linearly independent.

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91

(2) A row of Ak that has at least one zero has more than one but less than k one’s. For a fixed column of Ak , say its index is i for i ∈ {1, . . . , k},
rows of Ak that have a zero at position i and that have there are k−1 j Pk−1 k−1
= exactly j one’s for j = 2, 3, . . . , k−1. Hence, column i has j=2 j 2k−1 − (k − 1) − 1 = 2k−1 − k zero’s. Since Ak has 2k −k−1 rows each column contains 2k −k−1−(2k−1 −k) = 2k−1 − 1 one’s. (3) The proof is done by induction on k For k = 2 the statement is obvious. Consider the case k = 3: Define is(Ak ) to be the index set of the columns of A3 and is(I) to be the index set of the columns of I. Then, by (1), any circuit C of L3 can be written as C = X ∪ Y , where X ⊆ is(I) and Y ⊆ is(A3 ). Three cases have to be distinguished: |Y | = 1: Since every column of A3 has 3 one’s X must consist of the indices of 3 columns of I such that there are two one’s in every row of the column submatrix indexed by C. Hence the cardinality of C is equal to 4 = 23−1 . |Y | = 2: No two columns of Ak have all their one’s at the same positions and there is a single zero in every column. Hence a two column submatrix of A3 has two rows that have a single one. A circuit C containing the indices of these two columns must also contain the indices of two unit vectors of I such that the column submatrix indexed by C has two one’s in each row. Again, the cardinality of C is equal to 4 = 23−1 . |Y | = 3: One of the rows of A3 has an odd number of one’s. Hence, a circuit of L3 containing the indices of all 3 columns of A3 must also contain the index of one unit vector from I such that the mentioned row has an   even number of one’s in the column submatrix of I A3 indexed by C. Thus, |C| = 3 + 1 = 23−1 . Assume that the statement is true for all 2 ≤ n ≤ k.

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92

Inductive Step: Let Ck = Xk ∪ Yk be a circuit of Ak with Xk ⊆ is(I) and Yk ⊆ is(Ak ). Then |Ck | = |Xk | + |Yk | = 2k−1 . There are three possibilities to derive a circuit C of Ak+1 from Ck : Possibility 1: Choose the columns of Ak+1 that contain the |Yk | columns of Ak whose indices are in Ck . Then a circuit C of Lk+1 containing the indices of these columns must also contain the indices of 2 |Xk | + |Yk | distinct columns from I by the structure of Ak+1 . Hence, |C| = 2 · 2k−1 = 2k . s of these columns must also contain the indices of 2 |Xk | + |Yk | distinct columns from I by the structure of Ak+1 . Hence, |C| = 2 · 2k−1 = 2k . Possibility 2: Choose the columns of Ak+1 that contain the |Yk | columns of Ak whose indices are in Ck and the last column of Ak+1 . To obtain a minimal Eu  lerian column submatrix of I Ak+1 one has to choose |Xk | columns from I for the first 2k −k−1 rows and 2k −k−1−|Xk | other columns from I for the second 2k −k−1 rows as well as k−|Yk | other columns from I for the remaining rows. Hence |C| = 2k−1 +1+2k −k −1−|Xk |+k −|Yk | = 2k−1 + 2k − 2k−1 = 2k . Possibility 3: Choose the last column of Ak+1 and 2k − 1 appropriate columns from I. Then the indices of these columns form a circuit C of Lk+1 and |C| = 2k − 1 + 1 = 2k . (4) Let C be a circuit of Lk . Then 2 |C| = 2k > 2k − k − 1 + k = 2k − 1, i.e no distinct union of circuits is possible. (5) The number of one’s in a column of Ak is always less than the number of the rows from Ak for k ≥ 3. Hence, for any subset ∅ 6= Q ⊆ is(Ak ) there exists a circuit C of Lk that contains Q. Since ∅ is a cycle by definition and every nonempty cycle of Lk is actually a circuit Lk has


1 + k1 + k2 + . . . + kk = 2k cycles. The cycles of L2 are ∅, {1, 2}, {1, 3}, {2, 3}. Thus, the statement also holds for k = 2. 2

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93

Gr¨otschel and Truemper [11] used the following strategy to find an irredundant system describing P (Lk ). Since any Lk has no coloops and no coparallel classes P (Lk ) is full-dimensional by Theorem 5.12. Therefore, all facet defining inequalities describe P (Lk ) completely and irredundantly by Theorem 5.18. Thus, if a facet of P (Lk ) is found all other facets of P (Lk ) can be derived from that facet by applying Theorem 5.24. To handle all inequalities that have been obtained from a given inequality Gr¨otschel and Truemper [11] defined the following. Definition 5.37. Two inequalities are called related if one of them is obtained from the other by applying Theorem 5.24 to it. Clearly, this relation is an equivalence relation. Therefore, all facet defining inequalities form an equivalence class and any facet defining inequality is a representative of it. By the above results an inequality that is not even valid for P (Lk ) but also defines a facet of P (Lk ) can be found. This inequality is X

xi ≤ 2k−1 ,

(5.3.1)

i∈N

where N denotes the ground set of Lk . (5.3.1) is valid for P (Lk ) since every nonempty cycle C of Lk has cardinality 2k−1 . (5.3.1) is fulfilled with equality by all the 2k − 1 nonempty cycles of Lk . Since P (Lk ) is full-dimensional its dimension is 2k − 1. This implies that the k incidence vectors of the 2k cycles of P (Lk ) are affinely independent in R2 −1 . From this it follows that the incidence vectors of the 2k − 1 nonempty cycles k are linearly and thus affinely independent in R2 −1 . Hence, (5.3.1) defines a face of P (Lk ) of dimension 2k − 2 which is a facet by Theorem 5.15. From this inequality 2k − 1 other inequalities can be derived by applying Theorem 5.24 to the 2k − 1 nonempty cycles of Lk . Hence, the equivalence class of all facet defining inequalities of P (Lk ) consists of 2k elements. The following theorem summarizes the properties of complete binary matroids and gives a complete description of master polytopes.

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94

Theorem 5.38. For any k ≥ 1, the complete binary matroid Lk with ground set N has 2k − 1 elements and 2k − 1 nonempty cycles. Each nonempty cycle C is a circuit with |C| = 2k−1 . The cycle polytope P (Lk ) is full-dimensional and has 2k extreme points. The facet defining inequalities of P (Lk ) form a single equivalence class where X

xi ≤ 2k−1

i∈N

is a representative of it. In the following, cycle polytopes of matroids that can be obtained from complete binary matroids by deleting one element via contraction are investigated. If N is the ground set of the complete binary matroid Lk then any ˆ which is obtained by deleting one element via contraction can matroid M ˆ = Lk .T for T := N \ {e} and some e ∈ N . be written as M ˆ = Lk .T has no coloops and no coparallel classes It can be verified that M such as Lk . Consequently, all facet defining inequalities of P (Lk .T ) form an irredundant system that describes P (Lk .T ). Gr¨otschel and Truemper showed that Theorem 5.24 can be transformed such that, given a facet defining inequality of P (Lk .T ) and a circuit of Lk , it gives an other facet defining inequality of P (Lk .T ). The following theorem summarizes the results of Gr¨otschel and Truemper. For more details, [11] is referred. Theorem 5.39. Let k ≥ 2 and T := N \ {e}, where N is the ground set of Lk . Then the system of facet defining inequalities for P (Lk .T ) is represented by 2k−1 equivalence classes. Each of them corresponds to a circuit C of Lk that contains e and is given by the following system of inequalities X xi ≤ 2k−2 (5.3.2) i∈C /

X

i∈C∪D /

xi −

X

xi ≤ 0 for all circuits D of Lk with e ∈ / D.

(5.3.3)

i∈C∩D

ˆ = Lk .T is smaller than the ground set of Lk and the The ground set of M dimension of P (Lk .T ) is one less than the dimension of P (Lk ). Nevertheless, a minimal system describing P (Lk .T ) consists of much more inequalities than

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95

a system describing P (Lk ) irredundantly. In other words, the structure of P (Lk .T ) is more complex than that of P (Lk ). For example, P (L3 ) is completely described by 8 inequalities that form a P single equivalence class where i∈{1,...,7} xi ≤ 4 is a representative, see [2, 11] and Theorem 5.38. Consider the matroid L3 .{1, 3, 4, 5, 6, 7} represented by   1 0 0 1 1 0 A = 0 1 0 1 0 1 , 0 0 1 0 1 1

where the index set of the columns may be {1, 3, 4, 5, 6, 7}. In L3 the following four circuits {1, 2, 3, 5}, {1, 2, 4, 6}, {2, 3, 4, 7}, {2, 5, 6, 7} contain the index 2. Since all nonempty cycles of L3 are circuits and L3 has 8 cycles there are 3 circuits of L3 not containing the index 2. Therefore, a minimal system describing P (L3 .{1, 3, 4, 5, 6, 7}) is represented by 4 equivalence classes where each consists of 4 elements. Hence, the system has 16 inequalities, twice as many as a minimal system describing P (L3 ). L3 .{1, 3, 4, 5, 6, 7} has no F7∗ minor. Thus, it has the sum of circuits property. Since A is in standard form it follows by Proposition 3.44 that the rows of A are the incidence vectors of the fundamental cocircuits of L3 .{1, 3, 4, 5, 6, 7} corresponding to a certain base B and span the cocircuit space. Hence, any cocircuit can be written as symmetric difference of fundamental cocircuits by Theorem 3.32. Since in A every row has 3 one’s and every column has 1 or 2 one’s any pair of fundamental cocircuits has one element in common. Consequently, the set of cocircuits of L3 .{1, 3, 4, 5, 6, 7} contains 4 cocircuits of three elements and 3 cocircuits of four elements. All cocircuits of four elements are the symmetric difference of the 3 distinct pairs of fundamental cocircuits. Hence, only the cocircuits of three elements have no chord by Definition 5.32. Thus, a minimal system describing P (L3 .{1, 3, 4, 5, 6, 7}) is also given by the 16 cocircuit inequalities derived from the 4 cocircuits without chord, see Theorem 5.31. Note, that not all complete binary matroids that are contracted to a subset T have the sum of circuits property. There are two drawbacks of Theorem 5.39 concerning its computational use. First, the number of elements of Lk is exponential in the number of elements of Lk .T , in general. Second, the necessary operations to obtain the

5.4. Lifting facets of Master Polytopes

96

facet defining inequalities of P (Lk .T ) can not be done in polynomial time, in general, see [11].

5.4

Lifting facets of Master Polytopes

In this section, it becomes clear why the polytopes from the last section are called master polytopes. As it is shown next, facet defining inequalities of master polytopes also define facets of certain other binary matroids. The next theorem is a result of Gr¨otschel and Truemper. For a proof [11] is referred. ˆ be a binary matroid with ground set N Theorem 5.40. For k ≥ 2, let M   and standard representation matrix A = I R , where R is of the form X

Y

z

Ak

0

R=

(5.4.1) T

D

1

such that D is a proper row submatrix of  k  A . I ˆ by contracting it to N ˜ := N \ T first and by Then Lk can be obtained from M ˜ to N ˜ \ {z}. Moreover, each facet defining restricting the new matroid on N ˆ ), too. inequality of P (Lk ) defines a facet of P (M Note, that Theorem 5.40 gives some but, in general, not all facets of ˆ ). P (M

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For an example, consider the matrix A which represents the binary maˆ. troid M X 1 0 A= 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

T 0 0 0 0 1

1 1 1 0 1

Y 1 1 0 1 0

0 1 1 1 0

z 0 0 0 0 1

(5.4.2)

ˆ fulfills the conditions of Theorem 5.40 for k = 3. Indeed, the Clearly, M submatrix of A obtained by deleting the last row and the columns indexed by T ∪ {z} is a representation matrix for L3 . Hence, X ∪ Y is the ground set of L3 and the inequality X xi ≤ 4 i∈X∪Y

defines a facet of L3 by Theorem 5.38. The same inequality also defines a ˆ , by Theorem 5.40. In general, the indices in X ∪ Y are different facet of M ˆ it is in both matroids. (In L3 X ∪ Y is equal to {1, 2, 3, 4, 5, 6, 7} and in M equal to {1, 2, 3, 4, 6, 7, 8}.) Before the main result of [11] can be stated the following has to be defined. ˆ of a binary matroid M ˆ is called a complete Definition 5.41. A minor O ˆ is complete and can be obtained from M ˆ by contraccontraction minor if O ˆ is maximal with these properties it is called a tions only. If, additionally, O maximal complete contraction minor. The main theorem of [11] is the following. ˆ be a binary matroid without coloop and without coTheorem 5.42. Let M parallel elements. Suppose that the complete binary matroid Lk is a conˆ and let Nk be its ground set. Then the following are traction minor of M equivalent ˆ. (1) Lk is a maximal complete contraction minor of M ˆ ), too. (2) Every facet defining inequality of P (Lk ) defines a facet of P (M

5.4. Lifting facets of Master Polytopes

98

ˆ ). (3) At least one facet defining inequality of P (Lk ) defines a facet of P (M (4) The facet defining inequality ˆ ). facet of P (M

P

i∈Nk

xi ≤ 2k−1 of P (Lk ) also defines a

ˆ without coloops and without coparallel elements has always A matroid M L1 as a contraction minor. Indeed, it can be verified that contraction of all ˆ always yields L1 . up to one element of M For example, consider the binary matrix   1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1  (5.4.3) A= 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0

Clearly, there is no element in the row space of A that consists only of 1 or ˆ. 2 one’s and therefore there are no coloops and no coparallel elements in M
ˆ can be contracted such that A′ = 1 1 1 1 0 repBy Theorem 3.57 M ˆ , say M ˆ ’. Suppose that {1, 2, 3, 4, 5} is the resents a contraction minor of M ˆ ’. Then the independent sets of M ˆ ’ are ∅, {1}, {2}, {3}, {4}. ground set of M ˆ ’.{1} is the By Theorem and Definition 3.48 the only independent set of M
ˆ ’.{1} is given by A′′ = 0 . empty set. Thus, a representation matrix of M ˆ ”=M ˆ ’.{1} represented by A′′ consists of a single loop and is The matroid M ˆ is contained in therefore equal to L1 . In other words, the first element of M L1 . Obviously, there exists similar sequences of contraction that show that ˆ is contained in L1 . Consequently, any element of M ˆ is any element of M ˆ. contained in a maximal complete contraction minor of M If Theorem 5.42 should be used in practice one has to decide whether or not a given complete contraction minor is maximal. Gr¨otschel and Truemper [11] showed that an algorithm for this decision exists and that this algorithm runs in polynomial time. In the worst case this algorithm has to start with L1 ˆ without since the last is always a complete contraction minor of a matroid M coloops and without coparallel elements. In [11], Gr¨otschel and Truemper also showed how the maximal complete contraction minor Lk+l found by the algorithm can be constructed from the initial complete contraction minor Lk .

5.5. Cycle Polytopes of Binary Uniform Matroids Example 5.43. The representation matrix  1 1 0 0 1 0 1 0 1 0 0 1  1 0 0 1 0 0 B= 0 1 1 0 0 0  0 1 0 1 0 0 0 0 1 1 0 0

99

ˆ (K5 )∗ given by of M  0 0 0 0 0 0 0 0  1 0 0 0  0 1 0 0  0 0 1 0 0 0 0 1

,see Example 3.24, fulfills the assumptions of Theorem 5.42. Evidently, L2 is ˆ (K5 )∗ . Hence, any facet defining a maximal complete contraction minor of M ˆ (K5 )∗ ). inequality of L2 also defines a facet of P (M

5.5

Cycle Polytopes of Binary Uniform Matroids

Cycle polytopes of uniform matroids were studied by Gr¨otschel [9] who succeeded in describing the corresponding cycle polytopes completely. In this work, only binary matroids are of interest as it will become clear in Chapter 6. From Chapter 3, it is known that the class of uniform matroids is minor closed. By the Excluded Minor Theorem for Binary Matroids 3.62, a matroid is binary if and only if it has no U2,4 minor. Moreover, it can be verified by defining corresponding graphs that all matroids with 3 or less elements are graphic and therefore also binary. If these results a combined it turns out that only the uniform matroids U1,n for n ≥ 1 and Um,n for n ≤ 3 are binary. Since these matroids will not be considered in detail the systems describing their cycle polytopes are not given explicitly in this work. Fore more details, [9] is referred.

5.6

Cycle Problem of Binary Matroids

In this section, the cycle problem of binary matroids is introduced. The problem is a combinatorial optimization problem which consists of exponentially many constraints in general. Moreover, it turns out that the problem is NP-hard. First, an LP-relaxation for the cycle problem is given. Then it is investigated under which conditions this relaxation is polynomially solvable.

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100

The results of these investigations are used to classify binary matroids for which the cycle problem can be solved in polynomial time. This section summarizes the results of Gr¨otschel and Truemper published in [10]. The Cycle Problem of Binary Matroids is defined as follows. ˆ be a binary matroid. Then the combinatorial opDefinition 5.44. Let M timization problem ˆ} max{cT xC : C is a cycle of M

(5.6.1)

where c ∈ Rn defines the objective function is called the Cycle Problem of ˆ. M ˆ ) of M ˆ corresponds to Since any extreme point of the cycle polytope P (M ˆ and the optimal solution is always attained at an extreme point a cycle of M (5.6.1) is equivalent to ˆ )} mac{cT x : x ∈ P (M

(5.6.2)

Next, two examples for cycle problems are stated. ˆ is graphic then any cycle is the edge set of an Eulerian subgraph (1) If M of the corresponding graph G. In this case, the cycle problem is called the Eulerian subgraph problem. ˆ is cographic then any cycle forms an edge cut in a corresponding (2) If M graph H. In this case, the cycle problem is called the max-cut problem. It is well known that the max-cut problem is NP-hard, see [14]. Consequently, the cycle problem is also NP-hard in general since the max-cut problem is a special case of it. Recall from Section 5.2 that the trivial inequalities (5.2.1) and the cocirˆ ) of any binary cuit inequalities (5.2.2) are valid for the cycle polytope P (M ˆ . Also recall that P (M ˆ ) is always contained in the polytope matroid M ˆ ) := {x ∈ Rn : x fulfills (5.2.1) and (5.2.2)} Q(M

5.6. Cycle Problem of Binary Matroids

101

ˆ is binary. if M ˆ . Then, obviously, Suppose that N is the ground set of M ˆ ) = conv{x ∈ {0, 1}|N | : x ∈ Q(M ˆ )} P (M ˆ )} ≥ max{cT x : x ∈ P (M ˆ )} and the linear Hence, max{cT x : x ∈ Q(M program ˆ )} max{cT x : x ∈ Q(M (5.6.3) is an LP-relaxation of (5.6.2). Note, that a Linear Program is an optimization problem of the form max cT x Ax ≤ b x≥0 for c ∈ Rn , A ∈ Mat(m×n, R) and b ∈ Rm , see e.g. [12] and that the problem max{cT x : x ∈ Y ⊆ Rn } is an LP-relaxation of max{f (x) : x ∈ X ⊆ Rn } if X ⊆ Y and cT x ≥ f (x) for all x ∈ X, see e.g. [16]. Since the number of cocircuits may grow exponentially with the number ˆ the number of facets of Q(M ˆ ) may also grow exponentially. of elements of M Therefore, in general, it is not possible to write down all inequalities describˆ ) in polynomial time. ing Q(M In [10], Gr¨otschel and Truemper showed that the separation problem for ˆ ) can be solved in time polynomial in the size of the input if M ˆ is Q(M graphic, cographic or has no F7 , F7∗ minor. Gr¨otschel, Lov´asz and Schrijver [8] showed that an optimization problem can be solved in polynomial time if and only if the corresponding separation problem can be solved in polynomial time. Hence, it is possible to optimize ˆ ) if M ˆ is graphic, cographic or has no F7 , F7∗ minor. over Q(M ˆ in general. However, the optimal solutions do not correspond to cycles of M ˆ )=Q(M ˆ ) if and only if M ˆ has the sum of circuits property, see Since P (M Theorem 5.27, the following theorem can be formulated. ˆ is a binary matroid with the sum of circuits property Theorem 5.45. If M then the corresponding cycle problem can be solved in polynomial time.

5.6. Cycle Problem of Binary Matroids

102

ˆ (K5 )∗ and R10 is graphic and only Since none of the matroids F7∗ , M ˆ (K5 )∗ is cographic the following can be concluded. M Corollary 5.46. The cycle problem can be solved in polynomial time for ˆ (K5 )∗ minor. graphic matroids and for cographic matroids without M The cycle problem for graphic matroids was first solved by Edmonds and Johnson [5] using a matching algorithm. For cographic matroids ˆ (K5 )∗ minor, the corresponding cycle problem was first solved by without M Barahona [1]. ˆ ) can be solved in polynomial time The optimization problem over P (M for matroids with the sum of circuits property. However, it is not clear if there is an algorithm that solves the problem efficiently in practice. A complete description of Master Polytopes is known, see Theorem 5.38, but the number of inequalities may be exponential in the number of elements of ˆ . Consequently, it is to be expected that an algorithm for the correspondM ing optimization problem will not be efficient for practical use. Therefore, decomposition methods promise better polynomial time combinatorial algorithms for the cycle problem. Gr¨otschel and Truemper [10] developed such a decomposition method that is based on matroid decomposition. This method solves the cycle problem for matroids with the sum of circuits property and can even be extended to more general classes of matroids. In the last part of this section, the theoretical background for this method is presented, see [10]. ˆ be a binary matroid with the property that Theorem 5.47. Let M ˆ =M ˆ 1 ⊕e M ˆ 2 (M ˆ =M ˆ 1 ⊕Y M ˆ 2 ). Suppose that Alg1 is an algorithm for the M ˆ 1 and that Alg2 is an algorithm for the cycle problem of cycle problem of M ˆ 2 . Then the cycle problem M ˆ} max{cT xC : C is a cycle of M can be solved by running Alg1 two times (four times) and by running Alg2 once. Moreover, the encoding length of the objective functions for the cycle problems

5.6. Cycle Problem of Binary Matroids

103

ˆ 1 and M ˆ 2 are bounded by a polynomial in the that have to be solved for M ˆ. length of the objective function of the cycle problem for M Proof : Let N = {1, 2, . . . , n}, N1 = {1, 2, . . . , n1 } and N2 = {1, 2, . . . , n2 } ˆ, M ˆ 1 and M ˆ 2 , respectively. Let cT x with c ∈ Rn be be the ground set of M the objective function. Define M := (n + 1) · max{|ci | + 1, i ∈ N }. ˆ =M ˆ 1 ⊕e M ˆ 2. First suppose M In this case N1 ∩ N2 = {e}. Define two objective functions (c1,1 )T x and (c1,2 )T x by ( ( cj if j ∈ N1 \ {e} cj if j ∈ N1 \ {e} 1,2 1,1 and cj := cj := −M if j = e M if j = e ˆ 1 using Alg1. Denote the opSolve the corresponding cycle problems for M timal values by µ1,1 and µ1,2 and the optimal solutions by C1,1 and C1,2 , respectively. By the construction of c1,1 and c1,2 , e is contained in C1,1 but not in C1,2 . ˆ 2 by Define an objective function (c2 )T x for M ( ck if k ∈ N2 \ {e} c2k := µ1,1 − M − µ1,2 if k = e ˆ 2 using Alg2. Denote the optimal Solve the corresponding cycle problem for M value by µ2 and the optimal solution by C2 . ˆ is equal to µ1,2 + µ2 Then the optimal value µopt of the cycle problem for M and the optimal solution is ( (C1,1 ∪ C2 ) \ {e} if e ∈ C2 Copt := C1,2 ∪ C2 if e ∈ / C2 Indeed: µopt ≤ µ1,2 + µ2 : ˆ there exist a cycle D1 of M ˆ 1 and a cycle D2 of M ˆ 2 such For any cycle D of M that D1 ∆D2 = D and either D1 ∩ D2 = {e} or D1 ∩ D2 = ∅, see Section 4.7. Suppose that D1 ∩ D2 = ∅ then e is neither contained in D1 nor in D2 since D1 ∆D2 = D and X 1,2 X c T xD = cj + c2k ≤ µ1,2 + µ2 . j∈D1

k∈D2

5.6. Cycle Problem of Binary Matroids

104

Suppose that D1 ∩ D2 = {e} then X 1,1 X c2k − (µ1,1 − M − µ1,2 ) c T xD = cj − M + j∈D1

k∈D2

≤ µ1,1 − M + µ2 − µ1,1 + M + µ1,2 = µ1,2 + µ2 .

cT xopt = µ1,2 + µ2 :

cT xopt

( (c1,2 )T xC1,2 + (c2 )T xC2 if e ∈ / C2 = (c1,1 )T xC1,1 − M + (c2 )T xC2 − (µ1,1 − M − µ1,2 ) if e ∈ C2

In both cases cT xopt = µ1,2 + µ2 . ˆ =M ˆ 1 ⊕Y M ˆ 2: Suppose M In this case, N1 ∩ N2 = {r, s, t}. Define four objective functions (c1,1 )T x, (c1,2 )T x, (c1,3 )T x     if j ∈ N1 \ {r, s, t} if c j c j 1,1 1,2 cj := M , cj := M if j ∈ {r, s} if     −M if j = t −M if c1,3 j

  c j := M   −M

if j ∈ N1 \ {r, s, t} if j ∈ {s, t} if j = r

and

c1,4 j

( cj := −M

and (c1,4 )T x by j ∈ N1 \ {r, s, t} j ∈ {r, t} j=s

if j ∈ N1 \ {r, s, t} if j ∈ {r, s, t}

ˆ 1 using Alg1. Denote the optiSolve the corresponding cycle problems for M mal values by µr,s , µr,t , µs,t , µ ¯ and the optimal solutions by Cr,s , Cr,t , Cs,t , ¯ respectively. C, By the construction of c1,1 , c1,2 , c1,3 and c1,4 it holds: {r, s} ⊆ Cr,s , {r, t} ⊆ Cr,t , {s, t} ⊆ Cs,t and {r, s, t} ∩ C¯ = ∅. ˆ 2 by Define a objective function (c2 )T x for M  ck if k ∈ N2 \ {r, s, t}     (µr,t + µr,s − µs,t − µ ¯ − 2M )/2 if k = r c2k :=  ¯ − 2M )/2 if k = s  (µr,s + µs,t − µr,t − µ  (µr,t + µs,t − µr,s − µ ¯ − 2M )/2 if k = t

,

5.6. Cycle Problem of Binary Matroids

105

ˆ 2 using Alg2. Denote the optimal Solve the corresponding cycle problem for M value by µ2 and the optimal solution by C2 . ˆ is equal to µ Then the optimal value µopt of the cycle problem for M ¯ + µ2 and the optimal solution is  (Cr,s ∪ C2 ) \ {r, s} if r, s ∈ C2    (C ∪ C ) \ {r, t} if r, t ∈ C r,t 2 2 Copt :=  (Cs,t ∪ C2 ) \ {s, t} if s, t ∈ C2   ¯ C ∪ C2 if {r, s, t} ∩ C2 = ∅ The last can be proved analogously to the corresponding proof in part 1 using Proposition 4.22. ˆ 1 and M ˆ 2 it follows that By the construction of the objective functions for M their encoding lengths are bounded in the encoding length of the objective ˆ. function for M 2 In Chapter 6, a cycle problem is solved using the algorithm presented in the above proof. The following theorem stated by Gr¨otschel and Truemper [10] implicitly defines several classes of matroids for which the cycle problem can be solved in polynomial time. Theorem 5.48 (Gr¨ otschel and Truemper, see [10]). Let N be a class consisting of graphic matroids and of a finite number of matroids that are ˆ∈ non-graphic. Let M be a class of binary matroids such that for each M ˆ 1 ⊕e M ˆ 2 or a Y -sum M\N one can determine in polynomial time an e-sum M ˆ 1 ⊕Y M ˆ 2 for M ˆ , where M ˆ 1 ∈ N and M ˆ 2 ∈ M. Then there is a combinatorial M algorithm that solves the cycle problem for all matroids in M in polynomial time. Proof : The statement for the non-graphic matroids follows from Theorem 5.47 and induction. The statement for graphic matroids is clear since there exists a polynomial time algorithm for the cycle problem of graphic matroids. 2 This theorem allows to consider the cycle problem for binary matroids in a more general framework. Moreover, it states conditions that have to be

5.6. Cycle Problem of Binary Matroids

106

fulfilled by a class of matroids such that the cycle problem for all matroids contained in this class can be solved in polynomial time. For these reasons, the theorem is of fundamental importance for this work as it will turn out in Chapter 6. By Theorem 5.34 all matroids with the sum of circuits property can be decomposed via e- or Y -sum in graphic matroids or matroids isoˆ (K3,3 )∗ or M ˆ (V8 )∗ . From this section, it is known that the morphic to F7 , M cycle problem is polynomial time solvable for all matroids with the sum of circuits property. Hence, the choice of all graphic matroids and the matroids ˆ (K3,3 )∗ as well as M ˆ (V8 )∗ for N in Theorem 5.48 yields that M can be F7 , M one of the following five classes of binary matroids, see [10]. (1) the class of all graphic matroids (2) the class of all matroids with the sum of circuits property ˆ (K5 )∗ minor (3) all cographic matroids without M ˆ (K5 )∗ minor (4) all matroids without F7 and M ˆ (K5 )∗ minor (5) all matroids without F7∗ and M

Chapter 6 Codes and their relation to Matroids In this chapter, it is shown how binary linear codes and binary matroids are related. Moreover, the decomposition methods for binary matroids are transferred to binary linear codes. On the occasion, two new sums operating on generator matrices are introduced. In a second part, it is shown how the optimization results from Chapter 5 can be used for ML Decoding. Furthermore, several families of binary linear codes are defined for which ML Decoding can be done in polynomial time. It will turn out that these codes are good codes for ML Decoding but bad codes in the coding theoretical sense. The main part of this chapter follows the work of Kashyap [15] published in 2008.

6.1

Binary Matroids and Binary Linear Codes

From Chapter 2, it is known that any binary linear code C is uniquely defined by a parity check matrix H. By Definition 3.18, H represents the binary ˆ (H). Moreover, it is known that the circuit space of M ˆ (H) is the matroid M null space of the matrix H, see Proposition 3.38. By the definition of the parity check matrix, the null space of H is equal to the code C. Hence, every ˆ (H). codeword of C corresponds to the incidence vector of a cycle of M 107

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108

The following definition from Coding Theory enables to relate codewords of ˆ (H) more precisely. C and cycles of M Definition 6.1. Given a Code C, for every codeword c = (c1 , c2 , . . . , cn ) ∈ C its support is defined by supp(c) := {i : ci = 1}, i.e. the set of indices whose corresponding code bits are equal to one. A codeword 0 6= c ∈ C is called minimal if supp(d) * supp(c) for all d ∈ C, d 6= 0. In terms of the above definition the following can be observed. Observation 6.2. Let C be a binary linear code. Then 0 6= c ∈ C is minimal ˆ (H) represented by the if and only if supp(c) is a circuit of the matroid M parity check matrix H of the code C. Moreover, for any c ∈ {0, 1}n , c 6= 0 it ˆ (H). holds c ∈ C if and only if supp(c) is a cycle of M By Remark 3.19 different matrices might represent the same matroid. All these matrices can be obtained from each other by performing a sequence of the matrix operations listed in Remark 3.19. Obviously, these matrix operations also transform a parity check matrix in an other parity check matrix of the same binary linear code. The following proposition states this observation in other words, see [15]. ˆ (H) =M ˆ (H ′ ) Proposition 6.3. Let H and H ′ be binary matrices. Then M if and only if H and H ′ are parity check matrices of the same code. From this, it can be concluded that a unique binary matroid can be associated with each binary linear code and a unique binary linear code can be associated with each binary matroid. For example, the matroid 

 1 0 0 0 1 1 1 A = 0 1 0 1 0 1 1 0 0 1 1 1 0 1

that represents the Fano Matroid F7 is a parity check matrix for the [7, 4]-Hamming Code from Example 2.3. Hence, the [7, 4]-Hamming Code is

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109

equal to the circuit space of the Fano Matroid. In Coding Theory two binary linear codes C1 and C2 are called equivalent if there exists a permutation π of the coordinates of C2 such that C1 = π(C2 ) holds. In terms of Matroid Theory, this means that the binary linear codes C1 and C2 are equivalent if and only if the binary matroids associated with C1 and C2 are isomorphic, see Definition 3.14. The one to one correspondence between binary linear codes and binary matroids enables to applicate methods and results from Matroid Theory to Coding Theory. For this reason, the considerations in Chapter 5 are focused on binary matroids only. ˆ (H) is the matrix matroid represented by the parity check matrix If M ˆ (H) is the matroid M ˆ (G) H of the code C then the dual matroid of M represented by the generator matrix of the same code C. Indeed, the rows ˆ (H)∗ span the null space of H by Theorem of a representation matrix for M 3.21. By the definition of H, Gi · H T = 0 for every row Gi of G. More relations between binary matroids and binary linear codes are listed in the following table. All these relations can be concluded from the properties of binary matroids and binary linear codes.

6.1. Binary Matroids and Binary Linear Codes ˆ (H) Binary Matroid M cardinality of the ground set N ˆ (H)) rank(M ˆ (G)) rank(M a row of G any element of the null space of H ˆ (H) or a cocycle of a cycle of M ˆ (G) M ˆ (H) or a cocircuit of circuit of M ˆ (G) M ˆ (H) weight of a minimal circuit of M a row of H any element of the null space of G ˆ (G) or a cocycle of a cycle of M ˆ (H) M ˆ (G) or a cocircuit of circuit of M ˆ M (H) ˆ (G) weight of a minimal circuit of M ˆ (H)) cycle polytope P (M ˆ (G)) cycle polytope P (M

110

Binary Linear Code C code length n dim(C) dim(C ⊥ ) = n − dim(C) codeword of C codeword of C support of a codeword of C support of a minimal codeword of C minimal distance of C codeword of C ⊥ codeword of C ⊥ support of a codeword of C ⊥ support of a minimal codeword of C ⊥ minimal distance of C ⊥ codeword polytope P (C) codeword polytope P (C ⊥ )

The concepts of restriction and contraction of matroids can also be transferred to binary linear codes. Let H be a parity check matrix of the binary linear code C. Let N be ˆ (H) and let S, T be the ground set of the associated binary matroid M ˆ =M ˆ (H)|S.T of the matroid two disjoint subsets of N . Then, the minor O ˆ (H) can again be associated with a unique binary linear code. This code is M called a sub code of C and is denoted by C|S.T . In [15], restriction is called shortening and contraction is called puncturing. Shortening corresponds to deleting columns from a parity check matrix of C, see Remark 3.47, and puncturing corresponds to deleting columns in a generator matrix of C, see Proposition 3.57 and Theorem 3.21. ˆ can be associated with a binary linear code C As any binary matroid M the classification of binary matroids, see Figure 3.5 can also be considered as a classification of binary linear codes.

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111

By means of this classification, a graphic code is the binary linear code that can be associated with a graphic matroid. Definition 6.4. A binary linear code that has as a parity check matrix the vertex-edge incidence matrix of a graph G is called graphic code and is denoted by C(G). The class of all graphic codes is denoted by Γ. As a consequence of this association, all properties of graphic matroids can be transferred to graphic codes. Thus, the class of graphic codes is minor closed, see Proposition 3.54. Consequently, the class of graphic codes can be defined by an Excluded Minor Theorem as the graphic matroids, see Theorem 3.65. Theorem 6.5. A code is graphic if and only if it has no minor equivalent to the [7,4] Hamming Code or its dual or one of the codes C(K5 )⊥ and C(K3,3 )⊥ . In the same manner, one can define regular codes as follows. Definition 6.6. A binary linear code that has a totally unimodular parity check matrix is called regular. By Proposition 3.56, the class of regular codes is minor closed and can also be defined by the following excluded minor theorem, see Theorem 3.64. Theorem 6.7. A binary linear code is regular if and only if it does not contain as a minor any code equivalent to the [7,4]-Hamming-Code or its dual. Cographic and planar codes can be defined in the same way. Note, that this classification of binary linear codes is similar to the classification of binary matroids. Therefore, it can not be expected that this classification coincides with a coding theoretic classification of binary linear codes.

6.2

Composition Codes

and

Decomposition

of

In this section, the (de)composition methods introduced in Chapter 4 are transferred from binary matroids to binary linear codes. Additionally, an

6.2. Composition and Decomposition of Codes

112

equivalent approach to the (de)composition of binary linear codes, introduced by Kashyap [15], is presented. In this approach, the (de)compositions are constructed via generator matrices of given codes and not via the representation matrices of given matroids. Furthermore, the decomposition theorems for binary matroids, see Section 4.8, are formulated in terms of Coding Theory.

6.2.1

Direct Sum

Since any binary matroid can be associated with a unique binary linear code any direct sum of binary matroids can be considered as direct sum of binary linear codes. Consider Theorem and Definition 4.1 and let C1 and C2 be the binary linear ˆ 1 and M ˆ 2 , respectively. Then, the following propercodes associated with M ties can be deduced for the direct sum of binary linear codes. (1) For any codeword c of C1 ⊕ C2 there exists codewords c1 and c2 of C1 and C2 such that supp(c) = supp(c1 ) ∪ supp(c2 ). (2) For any codeword d of (C1 ⊕ C2 )⊥ there exists codewords d1 and d2 of C1⊥ and C2⊥ such that supp(d) = supp(d1 ) ∪ supp(d2 ). (3) (C1 ⊕ C2 )⊥ = C1⊥ ⊕ C2⊥ .

6.2.2

2- and 3-Sum

Definition 4.3 restated in the context of Coding Theory gives a definition of Tutte k-separations of binary linear codes. Note, that the rank of a matroid has to be replaced by the dimension of the associated code. Definition 6.8. Given a binary linear Code C of length n, for a positive integer k, a partition (T, T C ) of {1, 2, . . . , n} = N is called k-separation of C if

and

 min |T | , T C ≥ k

dim(C.T ) + dim(C.T C ) − dim(C) ≤ k − 1

(6.2.1) (6.2.2)

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113

The separation is called a minimal k-separation if equality holds in (1) and it is called an exact k-separation if equality holds in (2). All properties of Tutte k-connectedness, see Chapter 4, can also be transferred to binary linear codes. Hence, binary linear codes can also be classified according to Tutte k-connectedness, see Figure 4.1. Consequently, there exists also e-, ∆- and Y -sums of binary linear codes. A definition of the e-sum of binary linear codes is given by a reformulation of Definition 4.12 in the context of binary linear codes. Definition 6.9. Let C1 , C2 be two binary linear codes of length n1 ≥ 3, n2 ≥ 3, respectively. Let {e} be the unique common coordinate. Assume that there exists no codeword of weight 1 with its single one at coordinate e in C1 , C1⊥ , C2 or C2⊥ . Then, the binary linear code C with code length n = n1 + n2 − 2, which has the property that the support C of all codewords is of the form C = C1 ∆C2 where Ci is the support of a codeword of Ci for i = 1, 2, is an e-sum of C1 and C2 . The Investigations of Seymour [20] with regard to 2-sums can also be formulated for binary linear codes. Theorem 6.10. If (T, T C ) is an exact Tutte 2-separation of a binary linear code C then there exist a new coordinate e, a binary linear code C1 with bit numbers from T ∪ {e} and a binary linear code C2 with bit numbers from T C ∪ {e} such that C is an e-sum of C1 and C2 . Conversely, if C is an e-sum of C1 and C2 with bit numbers in N1 and N2 then (N1 \ N2 , N2 \ N1 ) is an exact Tutte 2-separation of C. Moreover, C1 and C2 are equivalent to proper minors of C. The theoretical consideration of 2-sums of binary linear codes is completed by a formulation of Theorem 4.24 in terms of Coding Theory. Theorem 6.11. Every binary linear code that is not Tutte 3-connected can be constructed from Tutte 3-connected proper sub codes of it by a sequence of direct sums and 2-sums.

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114

In a similar way, the results about ∆- and Y -sums can be transferred to binary linear codes. The corresponding theorems from Chapter 4 are not repeated in this section. However, the conditions for ∆- and Y -sums stated in Definition 4.14 and Definition 4.17 are explicitly formulated in coding theoretic language. Conditions for ∆-Sums: (∆1) C1 and C2 are binary linear codes with code lengths n1 , n2 ≥ 7. (∆2) C1 and C2 have three common coordinates, say e, f, g. (∆3) {e, f, g} is the support of a minimal codeword of C1 and C2 . (∆4) The support of every codeword of C1⊥ and C2⊥ is not contained in {e, f, g}. Conditions for Y -Sums: (Y 1) C1 and C2 are binary linear codes with code lengths n1 , n2 ≥ 7. (Y 2) C1 and C2 have three common coordinates, say r, s, t. (Y 3) {r, s, r} is the support of a minimal codeword of C1⊥ and C2⊥ . (Y 4) The support of every codeword of C1 and C2 is not contained in {r, s, t}. The properties of k-sums stated in Proposition 4.20 and Proposition 4.23 are also properties of e-, ∆- and Y -sums of binary linear codes. Hence, (C1 ⊕e C2 )⊥ = C1⊥ ⊕e C2⊥ , ⊥

C1⊥

⊕Y C2⊥

⊥

C1⊥

⊕∆ C2⊥ .

(C1 ⊕∆ C2 ) = (C1 ⊕Y C2 ) =

6.2.3

(6.2.3) and

(6.2.4) (6.2.5)

Equivalent 2- and 3-Sums

Since any binary linear code C can be associated with a unique binary ˆ the code C can be decomposed whenever M ˆ can be decomposed. matroid M The components of a decomposition, i.e matroids that are proper minors

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115

ˆ , can be associated with unique binary linear codes which are the of M components of the corresponding code decomposition. ˆ is the matroid represented by a Given the code C the associated matroid M ˆ and therefore parity check matrix H of C. In this approach the matroid M the code C is uniquely defined by the matrix H. From Coding Theory, it is known that the code C is also uniquely defined by a generator matrix G. Given a decomposition of a code in terms of matroids the corresponding representation matrices can be dualized from parity check matrices to generator matrices. A first question now is if C can also be decomposed in such a way that the components are directly given as generator matrices. In a second question, it can be asked if the generator matrices of two codes can be composed to a generator matrix of a code that is a sum of both codes. Such a code (de)composition was introduced by Kashyap [15]. It results from the analysis of the (de)composition of matroids described in Chapter 4. In this subsection, the (de)composition of Kashyap is derived from the results of Chapter 4. Furthermore, it is shown how both approaches are related. First, a relation between the parity check matrices and the generator matrices of a code for which an exact Tutte k-separation exists is introduced. Let C be a binary linear code with code length n and dimension r. Let (T, T C ) be an exact Tutte k-separation of C for k ≥ 1. Assume that dim(C|T ) = k1 and dim(C|T C ) = k2 then k1 +k2 −r = k −1 and by Proposition 4.16 there exists a parity check matrix H ∈ Mat(r × n, F2 ) of C that is of the form X1 H = Ik 1 0

X2 0 Ik2 −(k−1)

Y1 A1 D

Y2 0 A2

where T = X1 ∪ Y1 , T C = X2 ∪ Y2 and rank(D) = k − 1.

(6.2.6)

6.2. Composition and Decomposition of Codes

116

By Corollary 3.22 a G-Matrix of C has the form X1 G = AT1 0

X2 DT AT2

Y1 Ik 1 0

Y2 0

(6.2.7)

Ik2 −(k−1)

where rank(DT ) = rank(D) = k − 1. Thus, for any binary linear code C fulfilling the above assumptions there ˜ ∈ Mat(r × n, F2 ) given by exists a generator matrix G   Ik 1 E 0 F ˜ (6.2.8) G= 0 0 Ik2 −1 G where rank(F ) = k − 1 and a permutation π of the coordinates such that ˜ is a generator matrix of C. π(G) Next, it is shown how generator matrices can be composed.     Let G1 = g11 , . . . , gn1 1 , G2 = g12 , . . . , gn2 2 be generator matrices of the binary linear codes C1 , C2 with code lengths n1 , n2 . Let m be an integer such that 0 ≤ 2m < min{n1 , n2 }. Then the two generator matrices can be composed to a generator matrix of a new code with code length n = n1 + n2 − 2m by the following. First, construct the matrix  1  g1 . . . gn1 1 −m+1 gn1 1 −m . . . gn1 1 0 ... 0 ′ G := (6.2.9) 2 2 0 ... 0 g12 . . . gm gm+1 . . . gn2 2 Second, contract the m positions in G′ where G1 and G2 overlap. The resulting code is denoted by Sm (C1 , C2 ). In the following, the case k = 2 is considered and the corresponding 2-sum (de)composition is explained. According to the notation of Kashyap [15] this 2-sum is denoted by ⊕2 to distinguish it from the e-sum of matroids. Construction of ⊕2 -(De)Compositions The submatrix F of (6.2.8) has rank 1 for k = 2. Thus, there is at least one nonzero row and one nonzero column in F , call them f and f˜, respectively.

6.2. Composition and Decomposition of Codes

117

Furthermore, each row of F is either 0 or equal to f . f˜ is a column vector of length k1 whose ith component is 1 if the ith row of F is f and 0 otherwise. Hence, F = f˜ · f . The components G1 and G2 of a ⊕2 -decomposition are defined by     1 0 f ˜ (6.2.10) G1 := Ik1 E f and G2 := 0 Ik2 −1 G

If the code lengths n1 and n2 of the binary linear codes C1 and C2 generated by G1 and G2 are both greater than 3 a generator matrix of the code S1 (C1 , C2 ) can be constructed by contracting the column containing f˜ in (6.2.11)   Ik1 E f˜ 0 0  0 0 1 (6.2.11) 0 f  0 0 0 Ik2 −1 G which yields the matrix

G=



Ik 1 0

E 0

0 Ik2 −1

F G



(6.2.12)

since f˜ is a column vector of length k1 whose ith component is 1 if the ith row of F is f and 0 otherwise. During the composition the last coordinate of C1 and the first coordinate of C2 are eliminated. Since the ⊕2 -sum should be defined if and only if the e-sum is defined (0 . . . 01) must not be a codeword of C1 or C1⊥ and (10 . . . 0) must not be a codeword of C2 or C2⊥ by Definition 6.9. Therefore, the ⊕2 -sum can be defined as follows, see [15]. Definition 6.12. Let C1 , C2 be binary linear codes of length at least three, such that (0 . . . 01) is not a codeword of C1 or C1⊥ and (10 . . . 0) is not a codeword of C2 or C2⊥ . Then, S1 (C1 , C2 ) = C1 ⊕2 C2 . Since a detailed example for an e-sum (de)composition is given in Chapter 4 an example for a ⊕2 -sum (de)composition is cut out in this section. For more details, [15] is referred. Instead, it is explained how both sums are related. From the above results the following can be derived.

6.2. Composition and Decomposition of Codes

118

Observation 6.13. Let C1 , C2 be binary linear codes for which the e-sum can be defined for the associated matroids then the ⊕2 -sum can also be defined for the corresponding generator matrices and there exists a permutation π such that π(C1 ⊕2 C2 ) is the generator matrix of the code C1 ⊕e C2 . Conversely, if C is the code associated with the e-sum of the matroids associated with C1 and C2 then C is generated by the matrix π(C1 ⊕2 C2 ). In other words, the ⊕2 -sum is a sum of two generator matrices and the e-sum is the sum of two matroids but the components as well as the result of the e-sum are associated with unique binary linear codes. Both sums yield the same code C but, via the ⊕2 -sum, it is defined by a generator matrix G and, via the e-sum, it is defined by a parity check matrix H which is a representation matrix of the associated binary matroid. It is also possible to interpret the ⊕2 -sum in an other way. The code expressed by a ⊕2 -sum can be seen as the code associated with a matroid represented by the matrix G. In this case, the matroid associated with π(C1 ⊕2 C2 ) is the dual of the matroid associated with C1 ⊕e C2 . In terms of matroids, this can be written as ˆ (G1 ) ⊕2 M ˆ (G2 )) = (M ˆ (H1 ) ⊕e M ˆ (H2 ))∗ = (M ˆ (H1 ))∗ ⊕e (M ˆ (H2 ))∗ . π(M The last step follows from Proposition 4.20. Hence, by the definition of both ˆ (Gi ) and (M ˆ (Hi ))∗ are isomorphic for i = 1, 2. sums, the matroids M Besides the e-sum, the ⊕2 sum can be equivalently used for code (de)compositions of exact Tutte 2-connected codes. Thus, in a reformulation of the corresponding Decomposition Theorem 6.11 the 2-sum can be replaced by a ⊕2 - or e-sum of binary linear codes. As for the e-sum, Kashyap [15] introduced a code (de)composition that is equivalent to a ∆-sum (de)composition but that operates on G-matrices such as the ⊕2 -sum. The sum corresponding to this (de)composition is denoted by ⊕3 . The ⊕3 -sum is constructed in such a way that the last three and the first three coordinates are eliminated in the course of its performance.

6.2. Composition and Decomposition of Codes

119

Furthermore, it should be defined if and only if the ∆-sum for the matroids associated with the codes is defined. Thus, considering (∆1) - (∆4), the ⊕3 -sum can be defined as follows, see [15]. Definition 6.14. Let C1 , C2 be binary linear codes of length at least seven, such that (1) (0 . . . 0111) is a minimal codeword of C1 and in C1⊥ there exists no codeword whose support is entirely contained in the last three coordinates, except for the all zero codeword. (2) (1110 . . . 0) is a minimal codeword of C2 and in C2⊥ there exists no codeword whose support is entirely contained in the first three coordinates, except for the all zero codeword. Then, S3 (C1 , C2 ) =: C1 ⊕3 C2 . Conversely, a binary linear code C can be decomposed via a ⊕3 -sum if a ∆-sum decomposition of C is possible, i.e. if there exists an exact Tutte  3-separation (T, T C ) with min |T | , T C ≥ 4, see Theorem 4.15.

The submatrix F of (6.2.8) has rank 2 for k = 3. Thus, there are two linearly independent rows in F , call them x and y, such that x and y are a basis of the row space of F . Hence, any row in F is either 0, x, y or x + y. The components G1 and G2 of the ⊕3 -sum a constructed from (6.2.8), as follows. G1 is a binary (k1 + 1) × (|T | + 3) matrix defined by   Ik 1 E B (6.2.13) G1 := 0 0 111 where B ∈ Mat(k1 × 3, F2 )  000    001 and where the i th row of B is equal to  010    100

if if if if

the the the the

i i i i

th th th th

row row row row

of of of of

F F F F

is is is is

0 x y x+y

6.2. Composition and Decomposition of Codes G2 is a binary (k2 + 1) × ( T C + 3) matrix defined by     x+y I3 0 K where K :=  y  . G2 := 0 Ik2 −2 G x

120

(6.2.14)

Clearly, via the ⊕3 -sum, G1 and G2 can be composed to a generator matrix of C of the form (6.2.8). Thus, there exists a permutation π of the coordinates of C such that π(C1 ⊕3 C2 ) is a generator matrix for the code C1 ⊕∆ C2 , analogously to Observation 6.13.

In a similar way, Kashyap defined a sum equivalent to the Y -sum that fulfills (Y 1)-(Y 4) and again eliminates the last three coordinates of C1 and the first three coordinates of C2 . In [15], this sum which is denoted by ⊕3 is defined by the ⊕3 -sum. Before this definition is stated the relation between both 3-sums is revealed. If C1 and C2 are binary linear codes for which the ⊕3 -sum can be defined then (6.2.15) (C1 ⊕3 C2 )⊥ = C1⊥ ⊕3 C2⊥ . by Proposition 4.23. ˆ 1 ∗ and M ˆ 2 ∗ if it can be defined for Since the ∆-sum can not be defined for M ˆ 1 and M ˆ 2 the ⊕3 -sum can not be defined for codes C1⊥ and C2⊥ if it can be M defined for C1 and C2 . Hence, the ⊕3 -sum can only be applied to new codes C1⊥ and C2⊥ but not to C1⊥ and C2⊥ . More formally (C1 ⊕3 C2 )⊥ = C1⊥ ⊕3 C2⊥ = C1⊥ ⊕3 C2⊥

(6.2.16)

This allows to define the ⊕3 -sum by the ⊕3 -sum, see [15]. Definition 6.15. Let C1 , C2 be binary linear codes of length at least seven, such that

6.2. Composition and Decomposition of Codes

121

(1) (0 . . . 0111) is a minimal codeword of C1⊥ and in C1⊥ there exists no other codeword whose support is entirely contained in the last three coordinates, except for the all zero codeword. (2) (1110 . . . 0) is a minimal codeword of C2⊥ and in C2⊥ there exists no other codeword whose support is entirely contained in the first three coordinates, except for the all zero codeword. Then, the ⊕3 -sum of C1 and C2 is defined as C1 ⊕3 C2 = C1 ⊕3 C2 where C1 = C1 ∪ ((0 . . . 0111) + C1 ) and C2 = C2 ∪ ((1110 . . . 0) + C2 ). To show that the ⊕3 -sum is well defined it has to be proved that if the codes C1 and C2 fulfill the assumptions of Definition 6.14 then C1 and C2 fulfill the assumptions of Definition 6.15. ⊥ By the definition of C1 and C2 , (0 . . . 0111) ∈ C1 , (1110 . . . 0) ∈ C2 , C1 ⊂ C1⊥ ⊥ and C2 ⊂ C2⊥ . Hence, there is no nonzero codeword in C1 whose support is a proper subset of the last three coordinates and there is no nonzero codeword in C2 whose support is a proper subset of the first three coordinates. It remains to show that (0 . . . 0111) is a minimal codeword of C1 and (1110 . . . 0) is minimal codeword of C2 . Assume that there exists a codeword c ∈ C1 whose support is a proper subset of the last three coordinates. Then, by Definition 6.15 c must be contained in (0 . . . 0111) + C1 and c + (0 . . . 0111) ∈ C1 but this is impossible again by Definition 6.15. Analogously, it can be shown that (1110 . . . 0) is minimal in C2 . Consequently, any code that can be written as a ⊕3 -sum can also be written as ⊕3 -sum as shown in the next proposition. Proposition 6.16. Let C1 , C2 be binary linear codes for which the ⊕3 -sum can be defined. Then C1 ⊕3 C2 = (C1⊥ )⊥ ⊕3 (C2⊥ )⊥ . The statement can easily be proved by using Definition 6.15 and (6.2.15). As for the ⊕2 - and ⊕3 -sum, there exists a permutation π of the coordinates of C such that π(C1 ⊕3 C2 ) is a generator matrix for the code C1 ⊕Y C2 .

6.3. ML Decoding

122

By Theorem 4.25, all binary linear codes that are not internally Tutte 4connected can be decomposed via direct sum, ⊕2 -sum or ⊕3 -sum.

6.3

ML Decoding

As it is shown in Chapter 2, ML Decoding is equivalent to the minimization of a linear function over the set of all codewords. Since a binary linear code C with code length n is a finite subset of {0, 1}n the convex hull of all codewords of C is a polytope where any extreme point corresponds to a codeword of C, see Theorem 5.22. Thus, ML Decoding can be formulated as an LP. In Coding Theory, the convex hull in Rn of C ⊆ {0, 1}n is called the codeword polytope which will be denoted by P (C) in this work. By Definition 5.23 this polytope is equal to the cycle polytope corresponding to the matroid represented by any parity check matrix of C. Hence, ML Decoding is equivalent to solving the cycle problem transformed in a minimization problem for the associated binary matroid. Since the cycle problem is NP-hard ML Decoding is NP-hard as well. As explained above, ML Decoding is equivalent to the following LP. min{γ T x : x ∈ P (C)}

(6.3.1)

From the considerations in Section 5.6, it follows that min{γ T x : x ∈ Q(C)}

(6.3.2)

is an LP-relaxation of (6.3.1) where Q(C) is the polytope defined by the trivial inequalities (5.2.1) and the cocircuit inequalities (5.2.2). Recall, that a cocircuit of the matroid associated with C is the support of a minimal codeword of C ⊥ . In general, the description of Q(C) consists of exponential many inequalities since the number of codewords of C ⊥ may be exponential in the code length n. If the polytopes Q(C) and P (C) are equal the minimization problem over Q(C) (6.3.2) is equivalent to ML Decoding. Indeed, since C ⊆ Q(C) = P (C)

6.3. ML Decoding

123

the optimal solution is guaranteed to be an ML codeword. If Q(C) ⊂ P (C) the minimum can be attained at an extreme point of P (C) that is fractional. In this case, ML Decoding fails and the detected codeword is called a pseudocodeword. From Section 5.2 it is known that Q(C) = P (C) if and only if the matroid associated with C has the sum of circuits property. In terms of codes, a code has this property if and only if it has no sub codes equivalent to H7⊥ , C(K5 )⊥ or CR10 . By Theorem 5.45 ML Decoding can be done in polynomial time for these codes. Kashyap [15] called all codes whose associated matroids have the sum of circuits property geometrically perfect codes. The class of all geometrically perfect codes is denoted by G. This class can be defined by the following excluded minor theorem which is a reformulation of Theorem 5.34. Theorem 6.17. Any geometrically perfect code C can be constructed by a sequence of ⊕2 -sums and ⊕3 -sums from graphic codes and codes equivalent to H7 , C(K3,3 )⊥ or C(V8 )⊥ . By the results of Section 5.6 ML Decoding can be done in polynomial time for all codes contained in the following five families of binary linear codes. (1) the class of all graphic codes (2) the class of all geometrically perfect codes (3) all cographic codes without a sub code equivalent to C(K5 )⊥ (4) all codes without a sub code equivalent to H7 or C(K5 )⊥ (5) all codes without a sub code equivalent to H7⊥ or C(K5 )⊥ Let M be one of these five families of codes. Then there exists a family N that consists of all graphic codes and of finitely many non-graphic codes such that N and M fulfill the assumptions of Theorem 5.48 transferred to Coding Theory. Kashyap [15] called a family M of binary linear codes that fulfills Theorem

6.3. ML Decoding

124

5.48 for an appropriate family N of codes almost graphic. Thus, each of the above five families of codes is almost graphic. There might be more families of codes that are almost graphic and, hence, more codes for which ML Decoding can be done in polynomial time. All codes in such a family must have a decomposition via e- or Y -sum (⊕2 or ⊕3 -sum) by Theorem 5.48. However, no such family has been found yet. Moreover, it is not said that ML Decoding can only be done in polynomial time for almost graphic codes. Any complete decomposition of a code C in an almost graphic family M can be modeled by a decomposition tree with C as root and where any node has zero or two children. Clearly, any leaf, i.e. a node without any children, corresponds to a code that is graphic or that is one of finitely many non graphic codes, see [15]. In the following example ML Decoding is carried out for a real code using the algorithm given by the proof of Theorem 5.47. The corresponding optimization problem is solved in terms of matroid theory since, in this case, the Y -sum decomposition from Gr¨otschel and Truemper can be used. If the problem should be solved using a ⊕3 -sum decomposition this decomposition has to be derived from a ⊕3 -sum decomposition, first, using Proposition 6.16. Example 6.18. Let C be the extended parity check matrix  1 0 0 0 0 1 0 0 H= 0 0 1 0 0 0 0 1

[7, 4]-Hamming Code defined by the

1 0 1 1

1 1 0 1

1 1 1 0

 0 1 . 1 1

Let γ T x defined by γ T := [1, 20; 1, 09; 2, 28; −0, 15; 5, 08; −0, 98; −2, 85; 6, 20] be the objective function that has to be minimized. Note, that γ is chosen arbitrarily. By interchanging the 5th and 7th coordinate a code C˜ is obtained that is

6.3. ML Decoding

125

equivalent to C. C˜ is given by the  1 0 0 1 ˜ = H 0 0 0 0

parity check matrix  0 0 1 1 1 0 0 0 1 1 0 1  1 0 1 0 1 1 0 1 0 1 1 1

and γ T is transformed to γ˜ T = [1, 20; 1, 09; 2, 28; −0, 15; −2, 85; −0, 98; 5, 08; 6, 20]. ˆ (H). ˜ Let N = {1, 2, . . . , 8} be the ground set of the matroid M ˆ (H) ˜ with min{|T | , T C } ≥ 4 An exact Tutte 3-separation (T, T C ) for M is given by T = {1, 5, 6, 7} and T C = {2, 3, 4, 8}. Moreover, for X1 = {1}, ˜ is actually of the 3-sum X2 = {2, 3, 4}, Y1 = {5, 6, 7} and Y2 = {8} H decomposition form (4.3.1). ˆ (H) ˜ are represented by The components of a Y -sum decomposition of M X1 1 ˜ 1Y = 0 H 0 0

r 0 1 0 0

s 0 0 1 0

, see Section 4.3.2. The sub codes C1 , C2 codewords (0 0 0 (1 1 1 (1 1 0 (1 0 1 C1 : (0 0 1 (0 1 0 (0 1 1 (1 0 0

t 0 0 0 1

1 1 1 0

Y1 1 1 0 1

1 0 1 1

t 1 ˜ 2Y = 0 and H 0 0

0 1 0 0

X2 0 0 1 0

0 0 0 1

r 1 1 0 1

s Y2 1 0 0 1 1 1 1 1

ˆ (H ˜ 1Y ), M ˆ (H ˜ 2Y ) consist of the following associated with M 0 0 1 1 1 1 0 0

0 1 0 0 1 1 0 1

0 0 1 0 1 0 1 1

0), 0), 0), 1), 0), 1), 1), 1)

(0 (1 (1 (0 C2 : (0 (1 (1 (0

0 1 0 1 1 0 1 0

0 0 1 1 1 1 0 0

0 1 1 1 0 0 0 1

0 1 0 0 1 1 0 1

0 0 1 0 1 0 1 1

0), 0), 0), 1), 0), 1), 1), 1)

ˆ (H ˜ Y ) are given In the proof of Theorem 5.47 four objective functions for M 1 which, transformed in objective functions for the ML Decoding problem for C1 , are defined by (c1,1 )T := [˜ γ1 ; −M ; −M ; M ; γ˜5 ; γ˜6 ; γ˜7 ] = [1, 20; −M ; −M ; M ; −2, 85; −0, 98; 5, 08] (c1,2 )T := [˜ γ1 ; −M ; M ; −M ; γ˜5 ; γ˜6 ; γ˜7 ] = [1, 20; −M ; M ; −M ; −2, 85; −0, 98; 5, 08] (c1,3 )T := [˜ γ1 ; M ; −M ; −M ; γ˜5 ; γ˜6 ; γ˜7 ] = [1, 20; M ; −M ; −M ; −2, 85; −0, 98; 5, 08] (c1,4 )T := [˜ γ1 ; M ; M ; M ; γ˜5 ; γ˜6 ; γ˜7 ] = [1, 20; M ; M ; M ; −2, 85; −0, 98; 5, 08]

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126

The optimal values and optimal solutions of the corresponding cycle problems are, by straightforward checking µr,s = −1, 65−2M , µr,t = 0, 22−2M , µs,t = −3, 83−2M , µ ¯ = 0 and Cr,s = {1, r, s, 5}, Cr,t = {1, r, t, 6}, Cs,t = {s, t, 5, 6}, C¯ = ∅. ˆ (H ˜ Y ) is given by The objective function (c2 )T x for M 2 (c2 )T = [(µr,t + µs,t − µr,s − µ ¯ + 2M )/2; γ˜2 ; γ˜3 ; γ˜4 ; (µr,t + µr,s − µs,t − µ ¯ + 2M )/2; (µr,s + µs,t − µr,t − µ ¯ + 2M )/2; γ˜8 ] = [(˜ γ1 + γ˜6 − 2M + γ˜5 + γ˜6 − 2M − γ˜1 − γ˜5 + 4M )/2; γ˜2 ; . . . ; γ˜8 ] = [˜ γ6 ; γ˜2 ; γ˜3 ; γ˜4 ; γ˜1 ; γ˜8 ] = [−0, 98; 1, 09; 2, 28; −0, 15; 1, 20; −2, 85; 6, 20] By straightforward checking, the optimal value of the corresponding cycle problem is µ2 = −1, 7 and the optimal solution is C2 = {t, 3, 4, s}. ˆ (H) ˜ is µopt = µ Hence, the optimal value for M ¯ + µ2 = −1, 7 and the optimal solution is Copt = (Cs,t ∪ C2 ) \ {s, t} = {3, 4, 5, 6}. ˆ (H) the 5th and the 7th coordinate have To obtain the optimal solution for M to be changed again. Thus, {3, 4, 6, 7} is the support of the ML Codeword for the code C. As mentioned earlier, the binary linear codes defined according to the classification of binary matroids have not to be codes used in practice. Kashyap [15] even showed that almost graphic codes are not asymptotically good. That means that either the dimension or the minimal distance grow sub linearly with the code length n. Therefore, almost graphic codes are bad codes from the coding theoretical perspective.

6.4

Hamming Codes and Dual Hamming Codes

In this section, it is investigated if parts of the main results of this work can be applied to the classes of Hamming Codes and Dual Hamming Codes. All theorems, propositions and proofs are joint work with Akin Tanatmis. If the definition of Hamming Codes, see Example 2.3, and the definition of complete binary matroids, see Definition 5.35 are compared it turns out

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127

that, for k ≥ 3, any representation matrix of the dual matroid of some Lk is the parity check matrix of a Hamming Code for m = k. In other words, for k ≥ 3, the representation matrices of any Lk are parity check matrices of Dual Hamming Codes. Hence, the codeword polytopes of all Dual Hamming Codes are completely described by Theorem 5.38. From the definition of a complete binary matroid it follows that any Lk has an F7∗ contraction minor for k ≥ 3 since F7∗ is isomorphic to L3 . ˆ (K5 )∗ contraction minor for k ≥ 4 an M and an R10 contraction minor for k ≥ 5. From this it can be concluded that ML-Decoding can be done in polynomial time for k ≤ 3. For k > 3 the problem of ML-Decoding is no longer polynomially solvable since any Lk has minors isomorphic to both F7∗ and ˆ (K5 )∗ , see Section 5.6. M For m = 3, all Hamming Codes of length 7 are equivalent to H7 , i.e. the associated matroids are isomorphic to the Fano Matroid F7 . All these ˆ (K5 )∗ and R10 minor and, thus, they have the sum of matroids have no F7∗ , M circuits property. Hence, a complete description of the codeword polytope of all Hamming Codes of length 7 is given by Theorem 5.31. As a consequence of Theorem 5.34, Theorem 5.47 and Theorem 5.48 ML-Decoding can be done in polynomial time for all [7, 4]-Hamming Codes. For m ≥ 4, this property gets lost as stated in the next theorem. Theorem 6.19. Any Hamming Code of length n = 2m − 1, m ≥ 4 has a sub code equivalent to H7⊥ Proof : To prove the statement, it has to be shown that the matroid associated with any Hamming Code for m ≥ 4 has a minor which is isomorphic to F7∗ . Let Nm be the ground set of some Lm . Since a representation matrix of any matroid isomorphic to L∗4 has all distinct binary vectors of length 4 except for the zero vector as it columns a minor isomorphic to F7∗ can be obtained from any L∗4 by restrictions only. More formally, there exists a subset S ⊂ N4 such that F7∗ ∼ = L∗4 |S.

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128

Moreover, any Lm is a contraction minor of some Lm+1 for m ≥ 1. Thus, for all m ≥ 4 there exists a subset T ⊂ Nm+1 such that Lm ∼ = Lm+1 .T . By ∗ ∼ ∗ Theorem 3.49 the last is equivalent to Lm = Lm+1 |T . Together, it follows that for m ≥ 4 there exist a subset S ⊂ N4 and a subset V ⊂ Nm such that F7∗ ∼ = L∗4 |S ∼ = L∗m |V . Hence, for m ≥ 4, any matroid associated with a corresponding Hamming Code has a minor isomorphic to F7∗ . 2 From this theorem the following can be deduced. Proposition 6.20. For m ≥ 4, all Hamming Codes are not geometrically perfect codes. The question now is if the problem of ML-Decoding is polynomially solvable for Hamming Codes with m ≥ 4. Theorem 6.21. No Hamming Code can be decomposed via direct sum, 2sums or 3-sums. Proof : Let m ≥ 3 be a positive integer and let C be a corresponding [n, k]Hamming Code. By the definition of a Hamming Code, see Example 2.3, n = 2m − 1, k = 2m − m − 1 and there exists a parity check matrix H of C ˆ be the that has m rows and all the non zero m-tuples as its columns. Let M binary matroid represented by H and let N be its ground set. ˆ can not be decomposed via direct sum it suffices to show To prove that M ˆ is connected, see Lemma 4.11. Let e, f be two distinct elements of that M ˆ . Then the column vectors of H indexed by {e, f }, call them x, y, are M linearly independent by the structure of H. Again by the structure of H the vector z := x + y is a column of H, too. Let g by the index of this column. ˆ . Thus, M ˆ is connected by Definition Then the set {e, f, g} is a circuit of M 4.10. By the above argument every subset of N with cardinality two has rank two and a maximal subset of N with rank two has cardinality three. By an induction argument and the structure of H it follows that any subset of N with rank l has at most cardinality 2l − 1. ˆ can not be decomposed via 2-sums it is shown that the To prove that M

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129

assumption  min |T | , T C ≥ 2 and r(T ) + r(T C ) = m + 1 implies |T | + T C < n. From (6.4.1) it follows that r(T ), r(T C ) ≤ m − 1. Thus, C |T | + T C ≤ 2r(T ) − 1 + 2r(T ) − 1

(6.4.1)

≤ 2m−1 − 1 + 2m−1 − 1

= 2m − 2 < 2m − 1 = n

ˆ can not be decomposed via 3-sums is analog to that for The prove that M the 2-sums. 2 From the last theorem, it follows that, for m ≥ 4, all Hamming Codes are not almost graphic codes. Hence, for m ≥ 4, all Hamming Codes do not belong to the class of codes for which ML-Decoding can be done in polynomial time. Next, the associated codeword polytopes are considered. As mentioned above, the matroids associated with Dual Hamming Codes are isomorphic to complete binary matroids. By Theorem 5.38 a non empty cycle of any Lk is actually a circuit and has cardinality 2k−1 . Hence, there is no non zero codeword in any Dual Hamming Code with weight less or equal to 3. Thus, the associated matroid of any Hamming Code has no coloop, no coparallel elements and no triad. From this the following can be deduced. (1) The codeword polytope of any Hamming Code is full-dimensional. (2) L1 is a maximal complete contraction minor of a matroid associated with any Hamming Code. Thus, the inequalities 0 ≤ xe ≤ 1 define facets of the corresponding codeword polytope. Item (1) follows directly from Theorem 5.28. ˆ is contained in L1 and From Section 5.4 it is known that any element of M ˆ if M ˆ has no coloop and no coparallel that L1 is a contraction minor of M elements. If some L1 is contained in a triad then L2 is also a contraction

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130

ˆ and L1 is not maximal. These considerations together with minor of M Theorem 5.42 imply item (2). In a more general context, it is to be expected that many codes have dual codes with minimum distance greater than three since the number of errors that can be corrected grows with the minimum distance of a code. Hence, the codeword polytopes of all these codes are full-dimensional and the inequalities 0 ≤ xe ≤ 1 define facets of them.

Chapter 7 Conclusion 7.1

Summary

In this thesis, a particular optimization problem known as the problem of ML-Decoding in Coding Theory and known as the cycle problem in Matroid Theory is studied. This problem can be written as the minimization problem of a linear cost function over the codeword polytope which is the convex hull of all codewords. The same polytope is called cycle polytope in Matroid Theory where it is the convex hull of the incidence vectors of all cycles of a binary matroid. During the polyhedral analysis it turns out that a concise description of this polytope by means of inequalities increases exponentially in the code length. However, there are two known classes of binary matroids for which the corresponding cycle polytopes can be completely described. These classes are the matroids with the sum of circuits property and the complete binary matroids. A characterization of these matroids and the complete descriptions of the associated cycle polytopes are given in Chapter 5. A main result of the polyhedral analysis of cycle polytopes is that whenever an inequality that defines a face of a certain dimension is found some more inequalities that define faces of the same dimension can be deduced from the initial inequality. In Chapter 5, it is also shown how facet defining inequalities for the cycle polytope of a general binary matroid can be derived from facet defining inequalities of the cycle polytope of a complete binary matroid. Since there are exponentially many inequalities describing the cycle polytope of a binary matroid a combinatorial approach to the cycle problem is pro131

7.2. Future Research

132

posed. This approach uses matroid decomposition introduced by Seymour [20]. The theoretical background of matroid decomposition and concrete schemes for its construction are presented in Chapter 4. Starting from the decomposition of matroids the following strategy to solve the cycle problem is developed. First, decompose the matroid associated with the code in smaller matroids for which the corresponding cycle problem can be solved. Second, compose the optimal solutions of the “smaller” cycle problems to the optimal solution of the initial matroid. Gr¨otschel and Truemper [10] who developed and studied this approach to the cycle problem deduced a classification of matroids for which the cycle problem is polynomially solvable. In Chapter 6, it is presented how Kashyap [15] transferred the results of Gr¨otschel and Truemper from Matroid Theory to Coding Theory. In Section 6.3, the problem of ML-Decoding is solved for the extended [7, 4] Hamming Code and, in Section 6.4, it is shown that ML-Decoding can not be done in polynomial time for most Hamming Codes and Dual Hamming Codes.

7.2

Future Research

In a future research, the investigations about facet defining inequalities could be extended to other classes of binary linear codes. a first class could be the BCH-Codes since these codes are developed from Hamming Codes. An other possibility for a future research could be the search for new classes of almost graphic codes. Since a family C of binary linear codes is called almost graphic if there exists a finite sub class D ⊆ C of non-graphic codes such that any code in C can be decomposed via e- or Y -sum in graphic codes or in codes that are equivalent to the codes in D such D′ s have to be found. A nice tool for practical use would be an algorithm that decides whether or not a given matroid has a certain minor. Such an algorithm would not only be useful for the classification of matroids but also for the search for a complete contraction minor that is needed as starting point for the search for a maximal complete contraction minor, see Section 5.4. An algorithm that solves the problem of ML-Decoding for almost graphic codes would also be very nice. Such an algorithm would first have to gener-

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133

ate a decomposition tree, second, it would have to solve the cycle problems associated with the components of the decomposition and, last, it would have to compose the optimal solutions to the optimal solution of the initial MLDecoding problem. A final point for a future research could be the development of a construction scheme for the ⊕3 -sum introduced by Kashyap [15].

134

Bibliography [1] F. Barahona. The max-cut problem in graphs not contractible to K 5. Operations Research Letters, 2:107–111, 1983. [2] F. Barahona and M. Gr¨otschel. On the Cycle Polytope of a Binary Matroid. Journal of Combinatorial Theory (B), 40:40–62, 1986. [3] A. Beutelspacher. Lineare Algebra. Vieweg Verlag, Wiesbaden, 2003. [4] F. Bunke. Circuit Bases Problems in Binary Matroids. PhD thesis, Technische Universit¨at Kaiserslautern, Germany, 2006. [5] J. Edmonds and E. L. Johnson. Matching, euler tours and the Chinese postman. Math. Programming, 5:88–124, 1973. [6] J. Feldman. Decoding Error-Correcting Codes via Linear Programming. PhD thesis, Massachusetts Institute of Technology, USA, 2003. [7] Gerards B. Whittle G. Geelen, J. Towards a matroid-minor-structure theory. In G. Grimmet and C. McDiarmid, editors, Combinatorics, Complexity and Chance. A Tribute to Dominic Welsh. Oxford, U.K.:Oxford Univ. Press, 2007. [8] Lov´asz L. Schrijver A. Gr¨otschel, M. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:169–197, 1981. [9] M. Gr¨otschel. Cardinality Homogeneous Set Systems, Cycles in Matroids, and Associated Polytopes. Research Report 02-19, Konrad-ZuseZentrum f¨ ur Informationstechnik Berlin, 2002.

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[10] M Gr¨otschel and K. Truemper. Decomposition and Optimazation over Cycles in Binary Matroids. Journal of Combinatorial Theory (B), 46:306–337, 1989. [11] M. Gr¨otschel and K. Truemper. Master Polytopes for Cycles of Binary Matroids. Linear Algebra and its Applications, 114/115:523–540, 1989. [12] H. W. Hamacher and K. Klamroth. Lineare Optimierung und Netzwerkoptimierung. Vieweg, Wiesbaden, 2006. [13] D. Jungnickel. Codierungstheorie. Spektrum Akademischer Verlag, Heidelberg, Berlin, Oxford, 1995. [14] R. M. Karp. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thacher, editors, Complexity of Computer Computation, pages 85–103. Plenum Press, 1972. [15] N. Kashyap. A Decomposition Theory for Binary Linear Codes. IEEE Transactions on Information Theory, 54(7):3035–3058, 2008. [16] S. O. Krumke. Integer Programming. Lecture notes, http://www.mathematik.uni-kl.de/˜ krumke/lecturenotes.html. [17] S. Lin and D. J. Costello. Error Control Coding, Second Edition. Prentice-Hall, Upper Saddle River, NJ, USA, 2004. [18] J. G. Oxley. Matroid Theory. Oxford Science Publications. Oxford University Press, 1992. [19] N Robertson and P. D. Seymour. Generalizing Kuratowski’s Theorem. Congressus Numerantium, 45:129–138, 1984. [20] P. D. Seymour. Decomposition of Regular Matroids. Journal of Combinatorial Theory (B), 28:305–359, 1980. [21] P. D. Seymour. Matroids and multicommodity flows. European Journal of Combinatorics, 2:257–290, 1981. [22] K. Truemper. A Decomposition Theory for Matroids. I. General Results. Journal of Combinatorial Theory (B), 39:43–76, 1985.

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[23] W. T. Tutte. A homotopy theorem for matroids, I, II. Trans. Amer. Math. Soc., 88:144–174, 1958. [24] D. J. A. Welsh. Matroid Theory. L. M. S. Monographs. Academic Press, 1976. [25] H. Whitney. On the abstract properties of linear dependence. American Journal of Mathematics, 57:509–533, 1935.

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List of Figures 2.1

Basic Model of Coding Theory . . . . . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 3.5 3.6

Complete Graph K5 . . . . . . . . . . . . . . . Example for a disconnected graph . . . . . . . . A disconnected graph G . . . . . . . . . . . . . A connected graph H . . . . . . . . . . . . . . . Classification of Matroids . . . . . . . . . . . . The complete bipartite graph on three elements

. . . . . .

. . . . . .

12 13 21 21 23 37

4.1 4.2 4.3 4.4 4.5 4.6

Classification of matroids according to Tutte connectedness . Graphical interpretation of the direct sum of matroids (1) . Graphical interpretation of the direct sum of matroids (2) . Graphical interpretation of a 2-sum of matroids (1) . . . . . Graphical interpretation of a 2-sum of matroids (2) . . . . . Graphical interpretation of a 3-sum decomposition of a matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical interpretation of a Delta-sum decomposition of a matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical interpretation of a Y-sum decomposition of a matroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

43 54 55 56 58

4.7 4.8

5.1

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

2

. 59 . 61 . 62

The graph V8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

139

140

Declaration Neustadt, 22th Juli 2009

I hereby declare that I am the only author of this work and that no other sources than those listed have been used.

Sebastian Heupel

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