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Finding minimum 3-way cuts in hypergraphs ✩ Mingyu Xiao 1 University of Electronic Science and Technology of China, Chengdu 610054, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 18 October 2009 Received in revised form 29 April 2010 Accepted 5 May 2010 Available online 7 May 2010 Communicated by F.Y.L. Chin Keywords: Graph algorithms k-way cut Hypergraph

The minimum 3-way cut problem in an edge-weighted hypergraph is to find a partition of the vertices into 3 nonempty sets minimizing the total weight of hyperedges that have at least two endpoints in two different sets. In this paper we show that a minimum 3-way cut in hypergraphs can be found by using O (n3 ) hypergraph minimum (s, t ) cut computations, where n is the number of vertices in the hypergraph. Our simple algorithm is the first polynomial-time algorithm for finding minimum 3-way cuts in hypergraphs. © 2010 Elsevier B.V. All rights reserved.

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1. Introduction Given a hypergraph G = ( V , E ; w ) with nonnegative hyperedge weights, and an integer k, a k-way cut of G is a subset of hyperedges whose removal separates the hypergraph into k nonempty components, and the minimum k-way cut problem is to find a k-way cut minimizing the total weight of it. In the literature k-way cuts are also referred as k-cuts. The minimum k-way cut problem is an extension of the classical minimum cut problem, and has great applications in the area of VLSI system design, parallel computing systems, clustering, network reliability and finding cutting planes for the travelling salesman problems. The minimum k-way cut problem has been wellstudied in the literature. About twenty years ago Goldschmidt and Hochbaum [3] proved that the problem in ordinary graphs is NP-hard when k is part of the input and gave the first polynomial-time algorithm for fixed k. 2 Their algorithm runs in O (n(1/2−o(1))k ) time. Later, Kami-

✩ A preliminary version of this paper was presented at the 5th annual conference on Theory and Applications of Models of Computations (TAMC 2008). E-mail address: [email protected]. 1 Partially supported by National Natural Science Foundation of China Grant 60903007.

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doi et al. [6] gave an O (n4k/(1−1.71/ k )−31 )-time algorithm, and Xiao [21] improved the running time to O (n4k−log k ). Currently, the best results are the O (n2(k−1) log3 n)-time Monte Carlo algorithm due to Karger and Stein [8] and the O˜ (n2k )-time deterministic algorithm due to Thorup [19]. The minimum k-way cut problem has also drawn much attention in the literature for small values of k. The minimum 2-way cut problem is commonly known as the minimum cut problem. Another version of the minimum 2-way cut problem is the minimum (s, t ) cut problem, which asks us to find a minimum cut that separates two given vertices s and t. These two problems are classical and fundamental problems in the subject of graph connectivity. For ordinary graphs, the minimum cut problem can be solved in O (mn + n2 log n) time by Nagamochi and Ibaraki’s algorithm [14] or Stoer and Wagner’s algorithm [18], and the minimum (s, t ) cut problem can be solved in O (mn log n2 /m) time by Goldberg and Tarjan’s algorithm [2]. For hypergraphs, there are also some good results on these two problems. Klimmek and Wagner [10] and Mak and Wong [13] extended Stoer and Wagner’s algorithm [18] to hypergraphs and gave an O (dn + n2 log n)time algorithm for the minimum cut problem in hypergraphs, where d is the sum of the cardinalities of all hyperedges. Lawler [11] showed that a minimum (s, t ) cut in a hypergraph can be computed by using one maximum flow computation in an auxiliary digraph with n + 2m ver-

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tices and 2d + m edges. Then a minimum (s, t ) cut in a O (dm) time. In the remainder hypergraph can be found in  of the paper, we use T (n, m) and T h (n, m, d) to denote the running time of computing a minimum (s, t ) cut in an ordinary graph and a hypergraph respectively. In the paper, we will also consider the minimum ( S , T ) cuts that separate two disjoint nonempty vertex sets S and T . To find minimum ( S , T ) cuts, we can simply contract S into a single vertex s and contract T into a single vertex t, and then find minimum (s, t ) cuts in the graph. For the minimum 3-way cut problem in ordinary graphs, Kapoor [7] and Kamidoi et al. [5] showed that it can be solved by using O (n3 ) maximum flow computations. Burlet and Goldschmidt [1] and Nagamochi and Ibaraki [15] improved the result to O (n2 ) respectively. Furthermore, Nagamochi et al. [15,16] proved that the miniO (mnk ) time for mum k-way cut problem can be solved in  k = 4, 5, 6. Unfortunately, we do not find many results in hypergraphs for k  3. Currently the frequently used algorithms in VLSI system design to partition a hypergraph are some heuristic algorithms without any theoretic guarantee, such as algorithms based on the multilevel partitioning framework [9]. Effective algorithms for minimum 3-way cuts in hypergraphs have potential applications in VLSI system design and so on. A deep understanding of the structure of 3-way cuts will help us investigating the structure of the general k-way cut problem. Motivated by these, in this paper we study the minimum 3-way cut problem in hypergraphs. In the literature, there are three main techniques for designing algorithms for finding minimum 3-way cuts, as well as minimum k-way cuts, in ordinary graphs. The first technique is based on searching minimum (s, t ) cuts, such as the technique used by Goldschmidt and Hochbaum [3] to design the first nontrivial algorithm for the k-way cut problem. They proved that there exist four vertices a, b, c and d such that a minimum ({a, b}, {c , d}) cut is contained in a minimum 3-way cut. If we try all the O (n4 ) possibilities by taking each one as a subset of a 3-way cut and finding such a 3-way cut with minimum weight, then we can get a minimum 3-way cut by selecting a lightest one among all the O (n4 ) 3-way cuts. Later Kapoor [7] improved the running time to O (n3 ) maximum flow computations. The second technique is to enumerate all small 2-way cuts. We sort all cuts in the graph in the order of nondecreasing weights. If we try each cut as a subset of a 3-way cut, we will eventually meet a cut contained in a minimum 3-way cut and then get a minimum 3-way cut. Nagamochi et al. [15] proved that at least one of the first O (n) small cuts is contained in a minimum 3-way cut and the first O (n) cuts can be enunciated by using O (n2 ) maximum flow computations, implying that a minimum 3way cut can be found in O (n2 T (n, m)) time. Levine [12] proved that a minimum 3-way cut can be found by considering the first O (n) small cuts of all 4/3-minimum cuts (cuts with weight at most 4/3 times of the weight of a minimum cut) in three graphs. Since Karger and Stein’s algorithms [8] can find all 4/3-minimum cuts and a minimum cut in O (n2 log n) and O (m log3 n) time with high probability respectively, Levine got an O (mn log3 n)-time Monte Carlo algorithm for the minimum 3-way cut prob-

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lem. The last technique is based on a divide-and-conquer method, such as Kamidoi et al.’s algorithm [5]. We first find a proper cut of the graph that is noncrossing with a minimum 3-way cut, then we find minimum 3-way cuts in two smaller graphs obtained by contracting each part of the cut into a single vertex respectively. To get such a proper cut, Kamidoi et al. [5] used case analysis. All above properties and techniques in ordinary graphs cannot be extended to hypergraphs easily, because in contrast to the graph cut problems, it is difficult to find some cost bounding relationship between a minimum k-way cut and 2-way cuts in hypergraphs. In the initial version of this paper [20], we also showed an example that one of the fastest algorithms for finding minimum 3-way cuts in graphs does not work in hypergraphs. However the third technique may still be used to solve the k-way cut problem in hypergraphs, if we can find proper cuts noncrossing with a minimum k-way cut. In this paper, based on the Kamidoi et al.’s divide-and-conquer idea, we define a class of cuts called k-free cuts (see Definition 2.2) and present a framework of designing algorithms for the minimum kway cut problem in hypergraphs by finding k-free cuts. Then we prove that minimum 3-way cuts in hypergraphs can be found by using O (n3 ) hypergraph minimum (s, t ) cut computations. 2. Definitions and basic properties We refer to a hyperedge as a subset of vertices and use w (e ) to denote the weight of a hyperedge e ∈ E. Let X 1 , X 2 , . . . , X k ⊂ V be k (2  k  n) disjoint nonempty subsets of vertices, then [ X 1 , X 2 , . . . , X k ] denotes the set of hyperedges crossing at least two different vertex sets of { X 1 , X 2 , . . . , X k }. We also define another important hyperedge set: e ( X 1 , X 2 , . . . , X k ) = {e | e ∈ [ X 1 , X 2 , . . . , X k ] and k e ⊆ i =1 X i }. When { X 1 , X 2 , . . . , X k } is a k-partition of V , we say [ X 1 , X 2 , . . . , X k ] is a k-way cut of the hypergraph and X i is a component of the k-way cut. Over all k-way cuts those with the minimum weight are called minimum k-way cuts. For convenience, we will say a cut [ X , X ] is nontrivial if | X |  2 and | X |  2. Over all the nontrivial cuts those with minimum cost are called minimum nontrivial cuts. When no confusion arises, w ([ X 1 , X 2 , . . . , X k ]) is simply written as w ( X 1 , X 2 , . . . , X k ), and w (e ( X 1 , X 2 , . . . , X k )) as w e ( X 1 , X 2 , . . . , X k ). The following notion of noncrossing between a cut and a k-way cut plays a key role in our algorithms. We note that the property of noncrossing between two cuts are studied earlier [4]. Definition 2.1. A cut [ X , X ] and a k-way cut [Y 1 , Y 2 , . . . , Y k ] are noncrossing if there exists Y i (i ∈ {1, 2, . . . , k}) that entirely contains X or X . Otherwise, we say that they are crossing. Definition 2.2. A cut is called k-free if there exists a minimum k-way cut noncrossing with it. Lemma 2.3. Given a hypergraph G, let C = [ X , X ] be a cut of G, and C X and C X be a minimum k-way cut of G X and G X respectively, where G X is the hypergraph obtained by merging X into

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a single vertex and G X is the hypergraph obtained by merging X into a single vertex. If there is a minimum k-way cut noncrossing with cut C , then either C X or C X is a minimum k-way cut of G. Proof. Assume that minimum k-way cut C k = [Y 1 , Y 2 , . . . , Y k ] is noncrossing with cut C in G. Then there exists i ∈ {1, 2, . . . , k} such that X ⊆ Y i or X ⊆ Y i . If X ⊆ Y i , then w (C X )  w (C k ). If X ⊆ Y i , then w (C X )  w (C k ). Therefore, at least one of C X and C X is a minimum k-way cut. 2 Lemma 2.3 provides a potential recursive way to find a minimum k-way cut in a hypergraph: find a nontrivial cut that is noncrossing with a minimum k-way cut and then recursively find minimum k-way cuts in two smaller hypergraphs. This is one of the main ideas we will use to find minimum 3-way cuts in hypergraphs. We also need the following properties of hypergraphs, which can be easily proved by considering all possible hyperedges with given properties. Lemma 2.4. Let { X 1 , X 2 , . . . , X k } ∪ {Y 1 } be a group of disjoint nonempty subsets of vertices in a hypergraph, then

[ X 1 ∪ Y 1 , X 2 , . . . , Xk ] ⊇ [ X 1 , X 2 , . . . , Xk ] ∪ e (Y 1 , X 2 ∪ · · · ∪ Xk ).

Fig. 1. Illustration for Z i j and C 3 .

we prove that C 3 = [Y 1 ∪ Z 12 ∪ Z 13 , Z 22 , Z 23 ] is a minimum 3-way cut noncrossing with C . We only need to prove that w (C 3 )  w (C 3 ). Clearly, w ( Z 11 , Z 11 ) = w ( Z 11 , X 2 ) + w e ( Z 11 , Z 12 ∪ Z 13 ) and w ( X 1 , X 2 ) = w ( Z 11 , X 2 ) + w e ( Z 12 ∪ Z 13 , X 2 ). Since w ( X 1 , X 2 )  w ( Z 11 , Z 11 ), we have that

w e ( Z 12 ∪ Z 13 , X 2 )  w e ( Z 11 , Z 12 ∪ Z 13 ).

(1)

It is also easy to see that e ( Z 12 ∪ Z 13 , Z 22 ∪ Z 23 ) ⊆ e ( Z 12 ∪ Z 13 , X 2 ) and e ( Z 11 , Z 12 ∪ Z 13 ) ⊆ e (Y 1 , Z 12 ∪ Z 13 ). Then we have that

w e ( Z 12 ∪ Z 13 , Z 22 ∪ Z 23 )  w e (Y 1 , Z 12 ∪ Z 13 ).

(2)

If { X 1 , X 2 , . . . , X k } ∪ {Y 1 } is a partition of the vertex set, then

On the other hand, by Lemmas 2.4 and 2.5, we have that

[ X 1 ∪ Y 1 , X 2 , . . . , Xk ]

C 3 = [Y 1 ∪ Z 12 ∪ Z 13 , Z 22 , Z 23 ]

= [ X 1 , X 2 , . . . , Xk ] ∪ e (Y 1 , X 2 ∪ · · · ∪ Xk ).

= [Y 1 , Z 22 , Z 23 ] ∪ e ( Z 12 ∪ Z 13 , Z 22 ∪ Z 23 )

Lemma 2.5. Let { X 1 , X 2 , . . . , X k } ∪ {Y 1 , Y 2 } be a group of disjoint nonempty subsets of vertices in a hypergraph, then

and

[ X 1 ∪ Y 1 , X 2 ∪ Y 2 , X 3 , . . . , Xk ]

By (2), we get that w (C 3 )  w (Y 1 , Z 22 , Z 23 ) + w e ( Z 12 ∪ Z 13 , Z 22 ∪ Z 23 )  w (Y 1 , Z 22 , Z 23 ) + w e (Y 1 , Z 12 ∪ Z 13 )  w (C 3 ). Thus, C 3 is a minimum 3-way cut. 2

⊇ [ X 1 , . . . , Xk ] ∪ e (Y 1 ∪ Y 2 , X 3 ∪ · · · ∪ Xk ) ∪ e (Y 1 , X 2 ) ∪ e ( X 1 , Y 2 ). 3. Structures of 3-free cuts Recall that a 3-free cut is a cut that is noncrossing with a minimum 3-way cut. We lay the foundation of our algorithm by considering structures of 3-free cuts.

C 3 = [Y 1 , Y 2 , Y 3 ] ⊇ [Y 1 , Z 22 , Z 23 ] ∪ e (Y 1 , Z 12 ∪ Z 13 ).

Lemma 3.1 shows a noncrossing property between minimum cuts and minimum 3-way cuts. However, the noncrossing property between minimum (s, t ) cuts and minimum 3-way cuts is weaker. It is easy to construct examples that minimum (s, t ) cuts for two vertices s and t in an ordinary graph is not 3-free. But we still have the following lemma.

Lemma 3.1. In a hypergraph, any minimum cut is 3-free. Proof. We prove that for any minimum cut C = [ X 1 , X 2 ] of G, where X 2 = V − X 1 , there is a minimum 3-way cut C 3 such that C and C 3 are noncrossing. Let C 3 = [Y 1 , Y 2 , Y 3 ] be an arbitrary minimum 3-way cut, and Z i j = X i ∩ Y j , (i = 1, 2, j = 1, 2, 3) (see Fig. 1 for an illustration). If C and C 3 are crossing, then at least two of Z i1 , Z i2 , and Z i3 are nonempty (i = 1 or 2). Without loss of generality, we can assume that Z 11 , Z 22 and Z 23 are nonempty sets. (Note that we first assume Z 22 and Z 23 are nonempty. If Z 11 is empty, then Z 21 and Z 12 are nonempty. Therefore we can simply switch column Y 1 with column Y 2 .) Next,

Lemma 3.2. Given a hypergraph G and two disjoint nonempty vertex subsets S , T ⊂ V in G, if there is a minimum 3-way cut whose removal disconnects S from T (no component of the 3way cut contains both vertices in S and T ), then any minimum ( S , T ) cut is 3-free. Proof. The proof is based on the proof of Lemma 3.1. Let C 3 = [Y 1 , Y 2 , Y 3 ] be a minimum 3-way cut whose removal disconnects S from T , C = [ X 1 , X 2 ] a minimum ( S , T ) cut, and Z i j = X i ∩ Y j (i = 1, 2, j = 1, 2, 3). We will show that there is a minimum 3-way cut C 3 noncrossing with C . Since the removal of C 3 will disconnect S from T , either

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S or T is contained in one component of C 3 . Without loss of generality, we assume that S ⊆ Y 1 . Furthermore, we assume that S ⊆ X 1 and T ⊆ X 2 . Then S ⊆ Z 11 and T ⊆ Z 22 ∪ Z 23 . Note that [ Z 11 , Z 11 ] is still an ( S , T ) cut and then (1) still holds. If Z 22 and Z 23 are nonempty, the proof is exactly the proof of Lemma 3.1. Otherwise, we assume Z 22 = ∅ and T ⊆ Z 23 = ∅. For this case, Z 12 = ∅ and Z 11 = ∅. We switch column Y 1 with column Y 3 and switch column X 1 with column X 2 . Now T ∈ Z 11 and then (1) still holds. Furthermore, Z 22 and Z 23 = ∅. It also becomes the proof of Lemma 3.1. 2 Recall that a cut is nontrivial if each side of it has more than one vertex. If we want to use Lemma 2.3 to design algorithms for the minimum 3-way cut problem, we need to find a nontrivial 3-free cut [ X , X ]. If the 3-free cut is trivial, then G X or G X may be the same hypergraph as G. However, all 3-free cuts discussed above, including the minimum cuts and the minimum (s, t ) cuts, may be trivial cuts. Then we also need the following lemma. Lemma 3.3. Let G be a hypergraph with more than 6 vertices. If G has a minimum 3-way cut such that each component of it has at least 2 vertices, then any minimum nontrivial cut in G is 3-free. Proof. Let C 3

= [Y 1 , Y 2 , Y 3 ] be a minimum 3-way cut such that each component of it has at least two vertices, C = [ X 1 , X 2 ] an arbitrary minimum nontrivial cut, and Z i j = X i ∩ Y j (i = 1, 2, j = 1, 2, 3). We will prove the lemma by using Lemma 3.2. If there is a component of C 3 contained in a component of C , say Y 1 ⊆ X 1 , we can prove the lemma as follows. Let S = Y 1 and T = X 2 . Since both of S and T contain more than one vertex and C is a minimum nontrivial cut, we know C is a minimum ( S , T ) cut. Furthermore, the removal of C 3 will disconnect S from T . By Lemma 3.2, we know that the lemma holds. Otherwise, none of Z i j (i = 1, 2, j = 1, 2, 3) is an empty set. Since G has more than 6 vertices, at least one of Z i j (i = 1, 2, j = 1, 2, 3) contains more than one vertex. Without loss of generality, we assume that Z 11 contains at least two vertices. Let S = Z 11 and T = Z 22 ∪ Z 23 . Since none of Z 22 and Z 23 is an empty set, T also contains at least two vertices. Therefore, C is a minimum ( S , T ) cut. We can also prove this case by using Lemma 3.2. 2 4. A simple algorithm Now we use the structural results in Section 3 to design a polynomial-time algorithm for finding a minimum 3-way cut in a hypergraph. The idea is that: if there is a minimum 3-way cut such that one component of it has only one vertex, then we can find a minimum 3-way cut directly by taking each vertex as a component of a 3-way cut. Otherwise, the size of each component of a minimum 3-way cut is at least two. Then we find a nontrivial 3-free cut C = [ X , X ], and then recursively find minimum 3-way

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H3cut(G ) Input: An edge-weighted hypergraph G = ( V , E ; w ) with at least 3 vertices. Output: A minimum 3-way cut of G. 1. If the hypergraph has at most 6 vertices, return the solution and halt. 2. For each vertex v ∈ V , find a minimum cut C v in the induced sub hypergraph G [ V − { v }]. Let S be a 3-way cut with minimum weight among all the 3-way cuts C v ∪ [ v , V − { v }]. 3. Find a minimum nontrivial cut [ X , X ] in G, and return the best of S, H3cut(G X ) and H3cut(G X ). Fig. 2. The algorithm H3cut(G ). MinNontrivialCut(G ) Input: An edge-weighted hypergraph G = ( V , E ; w ) with at least 4 vertices. Output: A minimum nontrivial cut of G. 1. Arbitrarily select three different vertices a, b and c from V . 2. Return the best of MinTwo(G , {a, b}), MinTwo(G , {a, c }) and MinTwo(G , {b, c }). MinTwo(G , {x, y }) 1. For each vertex pair {u , v } ⊂ V − {x, y }, find a minimum ({x, y }, {u , v }) cut in G, and return a cut with minimum weight among all the minimum ({x, y }, {u , v }) cuts. Fig. 3. The algorithm MinNontrivialCut(G ).

cuts in the two smaller hypergraphs G X and G X . Our algorithm is given in Fig. 2. The correctness of the algorithm follows from Lemmas 2.3 and 3.3. Next, we focus on how to find a minimum nontrivial cut. Via brute-force search, a minimum nontrivial cut can be found by using O (n4 ) minimum (s, t ) cut computations. In fact, we can improve it to O (n2 ) minimum (s, t ) cut computations in the following way. Note that if there are four vertices a, b, c and d such that a minimum nontrivial cut separates {a, b} away from {c , d}, then any minimum ({a, b}, {c , d}) cut is a minimum nontrivial cut. Then for any three vertices, there exists a pair of vertices on a same side of a minimum nontrivial cut, and in the remaining n − 2 vertices, there exists a pair of vertices that are on the other side of the minimum nontrivial cut. If we arbitrarily select a set of three vertices and take each of the three vertex pairs in the set as a pair of vertices on one side of a minimum nontrivial cut, thenwe can find a minimum nontrivial cut by using 3×

n−2 2

= O (n2 ) minimum (s, t ) cut computations. The

algorithm is showed in Fig. 3. Now we are ready to analyze the running time of our algorithm H3cut. Suppose the current hypergraph has n vertices and | X | = x (2  x  n − 2). Then G X has x + 1 vertices and G X has n − x + 1 vertices. We get the recurrence relation:

 

C (n)  C (x + 1) + C (n − x + 1) + O n2 ,

(3)

where C (n) is the number of minimum (s, t ) cut computations used when H3cut is called for a hypergraph with n vertices, and O (n2 ) is taken by finding a minimum nontrivial cut. It is easy to verify that C (n) = O (n3 ) satisfies (3).

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Theorem 4.1. Algorithm H3cut finds a minimum 3-way cut O (dmn3 ) time, where in a hypergraph in O (n3 T h (n, m, d)) =  T h (n, m, d) is the running time of computing a minimum (s, t ) cut in a hypergraph. 5. Conclusion In this paper, we have presented the first polynomialtime algorithm for the minimum 3-way cut problem in hypergraphs. Recently, Okumoto et al. [17] proved that our algorithm can also be extended to solve the minimum 3way cut problem in submodular systems. They also gave some approximation algorithms for the minimum {4, 5}way cut problems in hypergraphs and submodular systems. But it still remains open whether the minimum k-way cut problem in hypergraphs for k  4 is NP-hard or not. k-way partitioning with minimum cost in hypergraphs is one of the central problems in VLSI system design. Effective algorithms for small values of k may have direct applications. Unfortunately, most properties and algorithms for k-way cuts in graphs do not hold in hypergraphs. Currently there are not many known algorithms for hypergraphs except some heuristic algorithms without any theoretic guarantee. It would be interesting to find other fast algorithms for the k-way cut problem in hypergraphs, even when k is a small number. References [1] M. Burlet, O. Goldschmidt, A new and improved algorithm for the 3-cut problem, Operations Res. Lett. 21 (5) (1997) 225–227. [2] A.V. Goldberg, R.E. Tarjan, A new approach to the maximum-flow problem, J. ACM 35 (4) (1988) 921–940. [3] O. Goldschmidt, D. Hochbaum, A polynomial algorithm for the k-cut problem for fixed k, Math. Operations Res. 19 (1) (1994) 24–37. A preliminary version appeared in FOCS 1988. [4] R.E. Gomory, T.C. Hu, Multi-terminal network flows, J. SIAM 9 (4) (1961) 551–670. [5] Y. Kamidoi, S. Wakabayashi, N. Yoshida, A divide-and-conquer approach to the minimum k-way cut problem, Algorithmica 32 (2) (2002) 262–276.

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