Intersecting Families, Independent Sets and Coloring of Certain Graph Products
Igor Shinkar
Advisor: Prof. Irit Dinur Department of Computer Science and Applied Mathematics Weizmann Institute of Science
i
Contents
I Intersecting Families and Independent Sets of Certain Graph Products 1 1 Introduction 1.1
Our results
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Preliminaries
5
7
2.1
Graph powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Pseudo-adjacency matrix and Homan's bound . . . . . . . . . . .
8
2.3
Boolean functions on [r]n . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Structure and stability of t-umvirates in [r]n
10
4 Intersecting families
14
4.1
Understanding the adjacency matrix of G . . . . . . . . . . . . . . .
15
4.2
The actual proof . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
5 General graph products
20
5.1
Main Theorem for general base graphs . . . . . . . . . . . . . . . .
21
5.2
Examples of graphs that satisfy the conditions of Theorem 5.1 . . .
26
5.3
A counter-example . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6 Concluding Remarks and Future Directions
29
II On the Conditional Hardness of Coloring a 4-colorable Graph with Super-Constant Number of Colors 33 7 Introduction
35
8 Preliminaries
37
8.1
Functions on the q -ary hypercube . . . . . . . . . . . . . . . . . . .
37
8.2
Functions in Gaussian space . . . . . . . . . . . . . . . . . . . . . .
40
8.3
The Majority is Stablest Theorem . . . . . . . . . . . . . . . . . . .
42
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9 A Variant of the Majority is Stablest Theorem 9.1
Proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Reduction
43 44
49
10.1 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
11 Conclusions and Final Remarks
52
A Minimum of Q(s)
61
iii
Abstract We study analytical methods applied to combinatorial problems. Speci cally, we use techniques from Fourier analysis in order to understand certain combinatorial structures. One of the main ideas of this approach is the following: Given a function of n variables, we wish to learn how a variable or a small subset of variables in uences the function, i.e. to understand how the change in a small number of variables aects the value of the function. For some problems the goal is to nd a small number of variables, that the given function essentially depends only on them. In other problems we are interested in nding a variable that has a non-negligible in uence on the function.
Part I
The rst part of this thesis deals with a variant of the intersection theorem. Given r ≥ 3 we are interested in understanding the maximal possible size of F ⊆ [r]n , a family of strings of length n over the alphabet [r] = {0, 1 . . . r − 1}, so that every two strings in F agree in at least t coordinates, for some xed value of t. Alternatively one can think of this problem as the problem of characterizing independent sets of maximal size in the corresponding intersection graph, G = (V, E). The vertices of the graph are V = [r]n , and two vertices are disconnected if and only if the corresponding strings equal in at least t coordinates. We show that for certain range of r and t, the relative size of any t-intersecting family F is at most 1/rt . The maximum can be attained by taking all strings with some xed t coordinates. These extremal examples are not only unique but also stable: For any ≥ 0 any such F of size 1/rt − can be approximated by some family de ned by xing exactly t coordinates.
Part II
In the second part we extend the result of Dinur et al. (STOC 2006) concerning the reduction from the 2-to-1 Label Cover problem to the Approximate Coloring problem. We show certain hardness of coloring a 4-colorable graph on n vertices with logc (n) colors (for some constant c > 0). This is done by proving a variant of the Majority is Stablest Theorem of Mossel at el. (FOCS 2005), which gives a tight bound on the noise stability of boolean functions in which every coordinate has small in uence.
iv
Acknowledgements I would like to thank my advisor, Prof. Irit Dinur, for her devoted guidance during the research. Her endless optimism and intuition have had an invaluable impact on this work. Also I am thankful to my fellow graduate students for creating a creative environment. It has been a pleasure to be part of this group for the past two years. I would like to thank my parents who taught me the value of education and hard work. They are one of the main reasons I am what I am today. I especially thank Eva for her support and patience during this di cult period.
v
Part I
Intersecting Families and Independent Sets of Certain Graph Products
1
Abstract We consider the following variant of an intersection theorem: Let F ⊆ [r]n be a family of strings such that every two strings in F agree in at least t coordinates. We show that for r ≥ t + 2 the relative size of F is at most 1/rt , which can be attained by taking all strings with some xed t coordinates. These extremal examples are not only unique but also stable: For any ≥ 0 any such F of size 1/rt − can be approximated by some family de ned by xing exactly t coordinates. We generalize this in the following sense: For a given a graph H = (V, E) de ne G to be a graph on vertices V n , where (x1 . . . xn ) and (y1 . . . yn ) are connected if (xi , yi ) ∈ E for all but at most t − 1 coordinates, that is (x1 . . . xn ) and (y1 . . . yn ) are not an edge if (xi , yi ) 6∈ E for at least t coordinates. We show that for certain graphs H the independence number α(G) of G equals α(H)t . By thinking of the edges as disagreement, the rst result is obtained as a special case by taking H to be a clique on r vertices. The proof idea is borrowed from Friedgut [Fri08], who uses spectral techniques applied on certain matrices that correspond to the relevant intersecting families. An important tool used in the proof is a theorem of Kindler and Safra [KS03], concerning boolean functions whose Fourier weight is concentrated on low levels.
2
1
Introduction
A classical problem in the extremal combinatorics is the problem of nding a maximal family A of k -subsets of [n] = {0 . . . n − 1} such that A ∩ B 6= ∅ for every
A, B ∈ A. More generally one could make a stronger requirement, that not only every two sets in A intersect, but for every A, B ∈ A holds |A ∩ B| ≥ t for some xed t ≥ 1. We say that such family A is t-intersecting. The study of nding the maximal t-intersecting families was initiated by Erdös, Ko and Rado. The following theorem was published in 1961 [EKR61], although, according to [Erd87], their work was essentially nished in 1938.
Theorem 1.1 (EKR) Let 1 ≤ t ≤ k. Then for n ≥ n0 (t, k) large enough |A| ≤
for any t-intersecting family A ∈
[n] k
n−t k−t
For t = 1 this bound is indeed the correct solution, and the only maximal intersecting families are the families of all sets containing some xed element. An extremely elegant proof of this fact was given by Katona [Kat72]. For general t ≥ 1 Frankl has made the following conjecture [Fra78]. De ne [n] n−t Ai = A ∈ : |A ∩ [1, t + 2i]| ≥ t + i 0≤i≤ 2 k Clearly each of the Ai is t-intersecting. Frankl conjectured that the t-intersecting families of maximal size of are isomorphic to some Ai , (i.e. equal to Ai up to a permutation of [n]). Following a long line of partial results (including [Fra78], [FF91], [Wil84]), Ahlswede and Khachatrian [AK97] con rmed the conjecture and the result is known as the Complete Intersection Theorem. We consider the following variant of the problem. Let F ⊆ [r]n be a family of strings over [r] = {0 . . . r − 1} of length n such that every two strings in F agree in at least t coordinates. We say that such F is t-intersecting. For xed
r ≥ 3 and t ≥ 1 we are interested in understanding the maximal possible size of F . One possible such family can be attained by taking all strings such that the rst
t coordinates equal 0. Another large family can be obtained by taking all strings that among t + 2 rst coordinates at least t + 1 of them equal 0. More generally 3
we de ne the following t-intersecting families for each i ≥ 0:
Fi = {x ∈ [r]n : at least t + i of x1 . . . xt+2i equal 0}
(1)
We say that a family F is isomorphic to Fi if it can be obtained from F by renaming the coordinates and (maybe) renaming the symbols in each coordinate. Formally there is a permutation of the coordinates σ ∈ Sn and permutations of the sym bols for each coordinate τ1 . . . τn ∈ Sr s.t. Fi = (τ1 (xσ(1) ) . . . τn (xσ(n) )) : x ∈ F . Frankl and Füredi [FF80] conjectured in 1980 that the only maximal t-intersecting families are isomorphic to some Fi (computing which of the Fi is the maximal is just an easy calculation). The special case of t = 1 has been proved independently by several dieren authors, e.g. [GL74],[Liv79],[ADFS04]. Ahlswede and Khachatrian [AK98] and Frankl and Tokushige [FT99] proved independently in 1998 that the t-intersecting families of maximal size of are isomorphic to some Fi , where i depends on r and t. Both proofs relied on the Complete Intersection Theorem [AK97] as a crucial building block. Ahlswede and Khachatrian call it the Diametric theorem:
Theorem 1.2 (Diametric theorem) Fix any natural
r ≥ 3 and t ≥ 2 and let
G ⊆ [r]n be a t-intersecting family of maximal size. Then
• If (r − 2) does not divide (t − 1), then G is isomorphic to Fb t−1 c . r−2
• If (r − 2) divides (t − 1), then G is isomorphic either to F t−1 −1 or to F t−1 . r−2
r−2
Alon et al. [ADFS04] gave a proof of Theorem 1.2 for the case of t = 1 using Fourier analysis on Znr . They showed that
|F | rn
≤
1 r
holds for any intersecting family
F and the maximal size is obtained only on families isomorphic to F0 (one can think of such families as dictators, i.e. deciding whether x = (x1 . . . xn ) belongs to the family depends on a single coordinate). The analytic method allows to study the dierence between dierent families, and, by using known results from Fourier analysis, deduce stability. It is shown in [ADFS04] that if F is a large intersecting family
|F | rn
=
1 r
− , then it can be approximated by a dictator, i.e. there is a
dictator G ⊆ [r]n s.t.
|F 4G| rn
< K, where K depends only on r. Ghandehar and
Hatami [GH08] improved this result by showing that in fact
|F 4G| rn
< C r for some
absolute constant C . Several variants of the intersection theorems are known. Gronau [Gro83] considered a slightly dierent setup of nding a maximal t-intersecting family F in the 4
product space [r1 ] × · · · × [rn ] for some xed r1 ≤ · · · ≤ rn . One can also consider the weighted version(see [FT03], [Tok07], [Kat64], [DS05], [Fri08]). For some xed the goal is to nd a t-intersecting family F ⊆ 2[n] of maximal weight, P where µp (F) = F ∈F p|F | (1 − p)n−|F | . For more related results see the excellent
0
1 2
survey of Bey and Engel [BE00]. For the weighted version Friedgut [Fri08] gave a proof that for certain range of
p and t, any t-intersecting family F ⊆ 2[n] of maximal size is de ned by exactly t coordinates. Moreover such families are stable: Any t-intersecting family of almost maximal size can be approximated by such family. The proof uses spectral techniques applied on certain matrices that correspond to the relevant intersecting families. An interesting part of the proof is that of de ning a pseudo-adjacency matrix of an appropriate graph over a ring that contains a nilpotent element, and not over the reals.
1.1 Our results In this paper we study the technique of Friedgut [Fri08] to extend the result of [ADFS04] for larger values of t and show stability for the special cases of r ≥ t + 2 of the Diametric theorem. We show that for r ≥ t + 2 the maximal t-intersecting family is obtained on some family F isomorphic to F0 , i.e. F is a family of all strings x ∈ [r]n that on some t coordinates (i1 . . . it ) equals some xed values (a1 . . . at ). That is, in order to decide whether x belongs to the family it is enough to compare some t coordinates with xed t values. We say that such family is a t-umvirate. Moreover, |F | ≥ 1/rt − , then rn |G4F | ≤ K, where K = rn
such sets are stable: If
there exists some t-umvirate family
G ⊆ [r] such that
K(r, t). The theorem is tight in the
n
sense that for r = t + 1 the maximum can be also attained by F1 , which cannot be approximated by a t-umvirate, and for r < t + 1 the maximum is not attained on F0 .
Theorem 1.3 (Main theorem) Let t-intersecting family. Then
t ≥ 2, r ≥ t + 2 and let F ⊆ [r]n be a
1. Upper Bound: µ(F) ≤ 1/rt . 2. Uniqueness: If µ(F) = 1/rt , then F is some t-umvirate. 5
3. Stability: If µ(F) ≥ 1/rt − , then there exists some t-umvirate family G ⊆ [r]n such that µ(G 4 F) ≤ K, where K = K(r, t). This is achieved by de ning a graph G = (V, E) whose vertices are [r]n , where two vertices are disconnected if they equal in at least t coordinates. It is clear that a t-intersecting family corresponds to an independent set in the graph. For t = 1 the graph G is exactly the n'th tensor product of an r-clique, that is the adjacency matrix of G is the tensor product (Kr )⊗n . In this case a tight bound on α(G), the independence number of G, can be obtained by Homan's bound ([Hof70],[Del73]), which we describe in the Section 2 (see also [ADFS04]). However for t ≥ 2 the graph seems to lack this nice property and Homan's bound is not tight anymore. The way we overcome this problem follows the idea of [Fri08]. Instead of considering the adjacency matrix of G over the reals, we de ne a pseudo-adjacency matrix of
Kr over a ring that contains a t-nilpotent element. Then, using spectral techniques on the matrix, we obtain the correct answer. We generalize the result stated above in the following way. For a xed graph
H = (V, E), de ne a graph G on vertices V n , where (x1 . . . xn ) and (y1 . . . yn ) are connected in G if and only if (xi , yi ) ∈ E for all but at most t − 1 coordinates. In other words (x1 . . . xn ) and (y1 . . . yn ) are not an edge if (xi , yi ) 6∈ E for at least t coordinates. If we think of the edges as disagreement, then two strings in G agree if they agree in at least t coordinates in H . We show that for certain graphs H the independence number α(G) of G equals α(H)t . We also give some indication that the independent sets of maximal size must have the structure of t-umvirate, however we are unable to show that. As a technical part of the proof we study the structure indicator functions of the t-umvirates. It is easy to see that if f : [r]n → {0, 1} is a t-umvirate, then
E[f ] = 1/rt and all its weight is concentrated on levels 0 through t of the Fourier transform. We show that in fact this is a characterization of such functions: If the weight of f : [r]n → {0, 1} is concentrated on levels 0 through t of its Fourier transform and E[f ] = 1/rt , then f is a t-umvirate. Using the powerful theorem of [KS03] we also prove stability of the theorem. The rest of the paper is organized as follows. Section 2 contains some basic de nitions including standard facts about Fourier analysis over Znr . In Section 3 we prove a technical theorem that allows us to understand t-umvirates in [r]n . 6
We show that any boolean function f with E[f ] = 1/rt whose Fourier weight is concentrated on levels 0 through t is a t-umvirate. In Section 4 we prove the main theorem. The proof is based on spectral techniques and makes use of the result from Section 3. Following that, in Section 5, we try to understand how the proof of the main theorem can be extended to general graphs. We achieve only partial results for general graphs. Finally in Section 6 we discuss open questions and possible future directions.
2
Preliminaries
We begin with some basic notation. Let r be some natural number. We de ne µr , the uniform measure on [r]n , in the natural way.
De nition 2.1 For F ⊆ [r]n , the measure of F is µr (F) =
|F| = Pr n [x ∈ F] x∈[r] rn
We will omit the subscript r as it is clear from context. It is immediate from the de nition that µ(F0 ) = 1/rt . We also need the following de nition.
De nition 2.2 A family
F ⊆ [r]n is called a t-umvirate if it is isomorphic to
F0 . That is there are indices 1 ≤ i1 . . . it ≤ n and a1 . . . at ∈ [r] s.t. F = {x ∈ [r]n : xi1 = a1 . . . xit = at }.
2.1 Graph powers Let H = (V, E) be a xed graph. The n'th tensor power of H , denoted by H ⊗n , is a graph on V n vertices (every vertex is a string of length n with symbols from V ), where two vertices (x1 . . . xn ) and (y1 . . . yn ) are connected if (xi , yi ) ∈ E for all i. For our purpose we need the following generalization:
De nition 2.3 For a given graph
t
H = (V, E), de ne H ⊗n to be a graph on V n
vertices. We put an edge between (x1 . . . xn ) and (y1 . . . yn ) if and only if (xi , yi ) ∈ E for all but at most t − 1 coordinates. In other words (x1 . . . xn ) and (y1 . . . yn ) are not connected if (xi , yi ) is not connected for at least some t coordinates. In order to explain the de nition we say two vertices x and y agree in H if and only if they are not connected by a edge. Then (x1 . . . xn ) and (y1 . . . yn ) agree in 7
t
H ⊗n if and only if (xi , yi ) agree in at least t coordinates. There is an obvious one to one correspondence between t-intersecting families in [r]n and independent sets t
of (Kr )⊗n . In Section 5 we will consider arbitrary graphs and try to understand t
the independent sets in H ⊗n .
2.2 Pseudo-adjacency matrix and Homan's bound Given a graph G = (V, E) on V = [r] vertices, we say that a matrix A ∈ RV ×V is its adjacency matrix if A(v, w) = 1 if (v, w) ∈ E and A(v, w) = 0 otherwise. The notion of pseudo-adjacency matrix is a relaxation of the above in the sense that the only requirement is that A(v, w) = 0 whenever (v, w) ∈ / E . We note that a pseudo-adjacency matrix need not be over the reals. In fact we will need to de ne a pseudo-adjacency matrix over a ring. As a motivation for this notion, let us now review Homan's bound for the independence number of a graph ([Hof70],[Del73], see also [GN08]). Let G =
([r], E) be a regular graph and let A be the normalized adjacency matrix of G. The eigenvalues of a symmetric matrix are real. Denote them by 1 = λ0 ≥ · · · ≥
λr−1 =: λ∞ , and let v0 = 1 . . . vr−1 be the corresponding orthonormal eigenvectors. We note that if f : V → {0, 1} is an indicator function of an independent set, then
f T r Af =
X
X
f (v)A(v, w)f (w) =
v,w∈V
f (v)A(v, w)f (w) = 0
(v,w)∈E
where the second equality is because A(v, w) = 0 for every (v, w) ∈ / E and the last equality follows from the fact that f is an indicator of an independent set. On the other hand, since {vi } form an orthonormal basis, f can be written as their linear P combination f = i ai vi , where ai = hf, vi i. Note that a0 = hf, 1i = E[f ] = α P 2 2 and by Parseval's identity r−1 i=0 ai = E[f ] = α. In this form we have
0 = hf, Af i =
X
λi a2i ≥ λ0 a20 + λr−1
i
r−1 X
a2i = λ0 α2 + λ∞ (α − α2 )
i=1
This implies Homan's bound:
Theorem 2.4 (Homan) Let G = (V, E) be a graph and let A be its symmetric pseudo-adjacency matrix with eigenvalues λ0 ≥ λ1 ≥ · · · ≥ λ|V |−1 =: λ∞ . Then α(G) ≤
−λ∞ λ0 − λ∞ 8
Moreover, if equality holds, then the indicating function of any maximal independent set is spanned by 1 and the eigenspace of λ∞ .
2.3 Boolean functions on [r]n For an element S = (S1 . . . Sn ) ∈ [r]n we denote supp(S) = {i : Si 6= 0} and |S| =
|supp(S)|. Consider the space of complex valued function with domain [r]n or, n
equivalently, a vector space Cr with inner product de ned as 1 X f (x)g(x) hf, gi = E[f g] = n r n x∈[r]
and norm of a vector de ned as
kf k2 =
p
hf, f i
For a ∈ [r] de ne the function χa to be
χa (x) = ω ax where ω = e
2πi r
is the primitive root of unity. It is a standard fact that {χa : a ∈ [r]}
is an orthonormal basis of complex functions on [r]. It de nes naturally an orn
thonormal basis of Cr by applying the n-fold tensor product. For S = (S1 . . . Sn ) ∈
[r]n de ne χS to be the function χS1 ⊗ · · · ⊗ χSn . More explicitly χS is a function from [r]n to C de ned by
χS (x) =
n Y
χSi (xi ) = ω hS,xi
i=1
where the inner product is modulo r. The set of functions {χS : S ∈ [r]n } forms an orthonormal basis of complex functions on [r]n . Thus any function f : [r]n → C can be written as
f=
X
fˆ(S)χS
where
fˆ(S) = hf, χS i
(2)
S∈[r]n
It will be convenient to think of the weight of a function at dierent levels. For
0 ≤ s ≤ n we de ne wf (s) =
X
|fˆ(S)|2
S:|S|=s
We usually omit the subscript in wf as it is clear from the context. The following proposition is standard and immediately follows from the de nitions. 9
Proposition 2.5 Let f : [r]n → C. Then E[f ] = fˆ(0)
E[|f |2 ] =
X
|fˆ(S)|2
S∈[r]n
In particular if f : [r]n → {0, 1} is a boolean function, then E[f ] =
X
|fˆ(S)|2 =
S∈[r]n
X
w(s)
s≥0
We state another useful proposition relating the Fourier coe cients of a function and its restrictions.
Proposition 2.6 Let
f : [r]n → {0, 1} and let g : [r]t → {0, 1} be a restric-
tion of f by xing the last n − t coordinates to zt+1 . . . zn , i.e. g(x1 . . . xt ) = f (x1 . . . xt , zt+1 . . . zn ). Then X
gˆ(T ) =
fˆ(T, S)χS (zt+1 . . . zn )
S∈[r]n−t
where fˆ(T, S) = fˆ(T1 . . . Tt , S1 . . . Sn−t ).
Proof
We write g in the Fourier basis decomposition
g(x1 . . . xt ) = f (x1 . . . xt , zt+1 . . . zn ) X fˆ(S)χS (x1 . . . xt , zt+1 . . . zn ) = S∈[r]n
=
X
X
T ∈[r]t
S∈[r]n−t
fˆ(T, S)χT (x1 . . . xt )χS (zt+1 . . . zn )
=
X
X
T ∈[r]t
fˆ(T, S)χS (zt+1 . . . zn ) χT (x1 . . . xt )
S∈[r]n−t
The proposition follows from the uniqueness of the orthonormal decomposition.
3
Structure and stability of t-umvirates in [r]n
In this section we study the t-umvirate functions. This understanding will be later used for the proof of the main theorem. It is easy to see that if f : [r]n → {0, 1} is a
t-umvirate, then E[f ] = 1/rt and all its weight is concentrated on levels 0 through 10
t of the Fourier transform. We show that in fact this is a characterization of such functions: If the weight of f : [r]n → {0, 1} is concentrated on levels 0 through t of its Fourier transform and E[f ] = 1/rt , then f is a t-umvirate. Using the powerful theorem of [KS03] we also prove stability of the theorem. In the theorem we assume that r ≥ 3. For r = 2 it is easy to show a counterexample. For instance consider the function f (x1 . . . xt+1 ) = AllEqual(x1 . . . xt+1 ) for any even t ≥ 2. Since f is an even function (f (x) = f (1 − x)), all its weight is concentrated only on even levels of the Fourier transform, in particular all the weight of fˆ is on levels 0 though t. In addition E[f ] = 21t and so the conditions of the theorem hold but the function depends on all t + 1 coordinates. This shows that the assumption of r ≥ 3 is necessary. We make the precise statement of the theorem below. The theorem corresponds to Lemma 2.8 of [Fri08]
Theorem 3.1 Let
t ≥ 2, r ≥ 3 and let f : [r]n → {0, 1} be a non-zero boolean
function s.t. E[f ] ≤ 1/rt . Then
1. Structure: If the weight of f is concentrated on levels 0 through t of its Fourier transform, i.e. X fˆ(S)χS
f=
S:|S|≤t
then f is a t-umvirate. 2. Stability: If f is has small weight above level t of its Fourier transform, i.e. for 0 < < 0 (r, t) X |fˆ(S)|2 ≤
S:|S|>t
then there is some t-umvirate g : [r]n → {0, 1} such that Prx [f (x) 6= g(x)] = kf − gk22 ≤ C, where C = C(r, t). In order to prove the theorem we need the following simple claim:
Claim 3.2 Let
ga , gb : [r]t → {0, 1} be two dierent t-umvirate functions. That
is ga = 1a and gb = 1b for a 6= b ∈ [r]t . Then there is S ∈ [r]t s.t. |S| = t and gˆa (S) 6= gˆb (S).
11
Proof
Assume w.l.o.g. that a1 6= b1 . In fact we prove a slightly stronger
statement: for any xing of (S2 . . . St ) there is S1 6= 0 s.t. gˆa (S) 6= gˆb (S), i.e. there are in fact many such S . The proof is simply by computing the Fourier coe cients of ga and gb and then comparing them.
1 −hS,ai ω rt Pn Then gˆa (S) = gˆb (S) i hS, ai = hS, bi mod r i (b1 − a1 )S1 = i=2 (ai − bi )Si gˆa (S) =
mod r. For any xing of (S2 . . . St ) the equality holds for at most half of the values of S1 (If r is prime then the equality holds for a unique value of S1 and otherwise there may be more solutions, but not more that 2r ). In particular, for r ≥ 3, there is S1 6= 0 s.t. hS, ai = 6 hS, bi and thus gˆa (S) 6= gˆb (S). We start the proof of the theorem by showing that if f depends on xi , then there is S ∈ supp(fˆ) s.t. |S| = t and Si 6= 0,
Claim 3.3 For every S
∈ [r]n s.t. |S| < t and fˆ(S) 6= 0 there is T extending S (that is Si = 6 0 implies Ti = Si ) s.t. |T | > |S| and fˆ(T ) 6= 0. In particular there is
such T extending S and |T | = t.
Proof
Assume w.l.o.g that supp(S) = {1 . . . j}, where j < t. Let g(x1 . . . xj ) be a
restriction of f so that E[g] ≤ E[f ] ≤ 1/rt . That is g(x1 . . . xj ) = f (x1 . . . xj , zj+1 , . . . , zn ) for some zj+1 , . . . , zn . Since g is a boolean function that depends on < t coordinates and E[g] ≤ 1/rt , we conclude that g ≡ 0 and in particular gˆ(S) = 0. On the other hand, using Proposition 2.6 we have X gˆ(S) = fˆ(S, 0) + fˆ(S, T )χT (zj+1 . . . zn ) T ∈[r]n−j \{0}
so there must be a non-zero element in the sum on the RHS.
e = t and Since f is not a constant function, we can x some Se such that |S| e 6= 0. We also may assume by rearranging the coordinates that supp(S) e = fˆ(S) {1 . . . t}. In order to prove the theorem we consider all possible restrictions of f to the last n−t coordinates and claim that all the restrictions de ne the same function. This will imply that f depends only on t coordinates and since E[f ] ≤ 1/rt this will imply that f is a t-umvirate.
12
Proof of Theorem 3.1 Structure:
De ne for each z = (zt+1 . . . zn ) a boolean
function hz on t coordinated in the natural way hz (x) = f (x, z) = f (x1 . . . xt , zt+1 . . . zn ). By Proposition 2.6 we have for every S ∈ [r]t and z ∈ [r]n−t X ˆ z (S) = fˆ(S) + fˆ(S, T )χT (zt+1 . . . zn ) h
(3)
T ∈[r]n−t \{0}
ˆ z (S) = fˆ(S) for every z . In particular hz is not a Observe that for |S| = t we have h ˆ z (S) e = fˆ(S) e 6= 0. Therefore, since hz depends on t coordinates zero function, as h we have E[hz ] ≥ 1/rt . On the other hand, the assumption Ex,z [f (x, z)] ≤ 1/rt leads us to the following inequality
1/rt ≤ Ez [Ex [hz (x)]] = Ex,z [f (x, z)] ≤ 1/rt implying that hz is a t-umvirate for every z , i.e. hz (x) = 1x=a (x) for some a ∈ [r]t . Now if we assume towards contradiction that hz 6≡ hz0 , then by the Claim 3.2 ˆ z (S 0 ) 6= h ˆ z0 (S 0 ). But this contradicts the equality there is S 0 s.t. |S 0 | = t and h ˆ z (S) = fˆ(S) for every z and |S| = t. This completes the proof of the theorem. h The proof of stability is similar to that appearing in [Fri08] using the powerful theorem of Kindler and Safra [KS03]. The theorem follows the line of results saying that if a function has small weight on the high coe cients or has small total in uence, then the function essentially depends on a small number of coordinates ([Bou02], [FKN02], [Fri98])
Theorem 3.4 (Kindler-Safra) Let r ≥ 3 and t ≥ 1 be two integers. Then there
are 0 > 0, C and K such that the following holds. Let < 0 and let f : [r]n → P {0, 1} be such that |S|>t fˆ(S) ≤ . Then there exists a function g : [r]n → {0, 1} that depends on at most K coordinates s.t. Prx [f (x) 6= g(x)] = kf − gk22 < C. Note that K , the number of in uencing coordinates in g , depends only on r and
t, and is independent of kg − f k, the proximity between f and g . In this sense it can be though of as a 'high-frequency FKN' [FKN02], that says that if a boolean function has most of its weight on the rst level of the Fourier transform, then it essentially depends on a single coordinate. Although the original theorem is stated for r = 2, one could adjust the original proof for our needs, essentially, by changing the basis. 13
Proof of Theorem 3.1 Stability:
2
We assume that kf >t k2 ≤ , for small
enough. By applying the Kindler-Safra theorem we get a function g that depends on K coordinates s.t. kf − gk22 < C. Clearly g 6≡ 0. It is enough to show that g is a t-umvirate. For this purpose de ne two constants. n o 2 γ = min g >t 2 : g : [r]K → {0, 1} s.t. g >t 6≡ 0 and
δ = min E[g] − 1/rt : g : [r]K → {0, 1} s.t. g >t ≡ 0, g is not zero and not a t-umvirate It isnclear that γo > 0. By the Structure item also δ > 0. Assume that < 2 γ , δC , We rst show that g >t ≡ 0: min 0 , √C+1
√
>t 2
√
g ≤ ( g >t − f >t + f >t )2 ≤ (kg − f k + )2 < ( C + 1)2 < γ 2 2 2 2 Therefore by the choice of γ , we conclude that g >t ≡ 0. Next we show that g is a
t-umvirate: (E[g] − E[f ])2 = (ˆ g (0) − fˆ(0))2 ≤ kg − f k22 < C < δ 2 Therefore E[g] < 1/rt + δ and by the choice of δ we conclude that g is a t-umvirate as required.
4
Intersecting families
In this section we prove the main theorem. We only consider t-intersecting families for t ≥ 2. Theorem 1.2 states that for the only maximal t-intersecting families are the t-umvirates. For r = t + 1 the maximum is also attained by 't + 1 out of t + 2' type of family, which cannot be approximated by a t-umvirate, and for r < t + 1 the maximum is not attained on a t-umvirate. For the range of r ≥ t + 2 we prove not only the uniqueness but also the stability in the following theorem.
Theorem 4.1 Let Then
t ≥ 2, r ≥ t + 2 and let F ⊆ [r]n be a t-intersecting family.
1. Upper Bound: µ(F) ≤ 1/rt . 2. Uniqueness: If µ(F) = 1/rt , then F is some t-umvirate. 14
3. Stability: If µ(F) ≥ 1/rt − , then there exists some t-umvirate family G ⊆ [r]n such that µ(G 4 F) ≤ K, where K = K(r, t). As discussed previously we consider a graph G on vertices V = [r]n , where
(x1 . . . xn ) and (y1 . . . yn ) are connected if and only if xi 6= yi for all but at most t coordinates. There is a clear one to one correspondence between t-intersecting families in [r]n and independent sets of G. Thus our goal is to show that α(G) ≤ 1/rt and any large independent set of G has the nice structure of t-umvirate families.
4.1 Understanding the adjacency matrix of G Before we proceed with the actual proof, let's rst try to understand the adjacency matrix of G. In order to do that we compute the spectrum of G. This is done by imitating the idea of Friedgut [Fri08] of working over a certain ring of polynomials and not over the reals. We will then see that Homan's bound is not tight for the adjacency matrix of G. We express BG , the adjacency matrix of G, as follows: Let X be a formal variable. Later we will replace it by an actual value to get the adjacency matrix. De ne B = Kr + XIr where Kr = (Jr − Ir ) is the adjacency matrix of an r-clique, i.e.
X .. . .. .
1 ..
1 P
...
B= For v ⊥ 1 we have (Bv)i = Xvi +
j6=i
... .
1
.. . 1 1 X
vj = Xvi − vi . Thus the matrix B has two
eigenvalues: X + r − 1 and X − 1
B1 = (X + r − 1)1 Bv = (X − 1)v
for v ⊥ 1
Consider the matrix B ⊗n . In this matrix if x and y agree in k coordinates then
B ⊗n (x, y) = X k The eigenvalues of B ⊗n are the products of the eigenvalues of B . λs = (X − 1)s (X + r − 1)n−s
s = 0...n
The adjacency matrix BG is obtained from B ⊗n by rst setting X t = 0 and then letting X = 1. We stress that this is only a syntactic substitution and it is easy to check that the result is indeed the required adjacency matrix. 15
We rst compute λs modulo X t = 0: s
(X − 1) =
t−1 X s i=0
n−s
(X + r − 1)
=
t−1 X n−s j
j=0
(where
s i
i
(−1)s−i X i
(r − 1)n−s−j X j
= 0 for s < i). Multiplying the two terms modulo X t = 0 gives us ! t−1 k X X s n−s λs = (−1)s−k+j (r − 1)n−s−j X k k − j j j=0 k=0
Then by setting X = 1 we get the spectrum of BG :
t−1 X k X s n−s λs = (−1)s−k+j (r − 1)n−s−j k − j j k=0 j=0
s = 0...n
For example the eigenvalue corresponding to the eigenvector 1 is λ0 =
Pt−1
k=0
n k
(r−
1)n−k . Next we show that applying Homan's bound on BG will not give the correct value of the independence number. For simplicity we x t = 2 (larger values of t can be handled similarly). It is easy to check that n−s s n−s λs = (−1) (r − 1) +1−s s = 0...n r−1 n In particular λ0 = (r − 1)n r−1 + 1 and the smallest eigenvalue is λ1 = −(r − 1 1)n−1 n−1 . Applying Homan's bound gives us α(G) ≤ λ−λ = 1−λ10 /λ1 = r−1 0 −λ1 1 . 1+(r−1)[ n+r−1 ] n−1
For n large enough compared to r the bound gives essentially
α(G) ≤ 1/r and not α(G) ≤ 1/r2 . This example shows that this bound is not always useful "out of the box". However a slight sophistication overcomes this in the next subsection.
4.2 The actual proof We start the proof by de ning a pseudo-adjacency matrix AG of G. The correct choice of AG is essentially the most important part of the proof. Once we de ne it, the rest of the proof is just a technical issue. De ne an r × r matrix A over the ring R[X]/(X t = 0) to be A = (1 − X)Kr + XIr , where Kr = 16
1 (Jr r−1
− Ir ) is
the normalized adjacency matrix of an r-clique (unlike in the previous subsection, here we normalize the matrix, though this is not really necessary and is done only for convenience and compatibility with the general case in the next section). 1−X X 1−X . . . r−1 r−1 . .. .. . A= . . .. . . 1−X r−1 1−X 1−X . . . X r−1 r−1 Observe that A has two eigenvalues: 1 and and
A1 = 1
rX−1 r−1
Av =
rX − 1 v for v ⊥ 1 r−1
We x an eigenbasis of A to be {χ0 = 1, χ1 . . . χr−1 } (think of χi as a column vector χi = (1, ω i , ω 2i . . . ω (r−1)i )0 ), and the corresponding eigenvalues are λ0 = 1 and λj =
rX−1 r−1
for j 6= 0. We de ne the pseudo-adjacency matrix of G to be
AG = A⊗n More explicitly if x and y agree in k coordinates then AG (x, y) = X k
1−X n−k r−1 t
One
checks easily that indeed AG is a pseudo-adjacency matrix of G (as X = 0). Using standard facts from linear algebra, the eigenvectors of AG are χS = χS1 ⊗ · · · ⊗ χSn and the corresponding eigenvalues are |S| n Y 1 λS = λ Si = − (1 − rX)|S| r−1 i=1 Using the binomial formula we compute the coe cients of X j in (1 − rX)|S| for
j = 0 . . . t − 1 which in turn gives us the following proposition.
Proposition 4.2 For S ∈ [r]n of size |S| = s the eigenvalue corresponding to the eigenvector χS is
λs =
(If s < j ,
s j
1 − r−1
s
! s (−1)j rj X j j j=0
t−1 X
is zero.)
Now let F ⊆ [r]n be an independent set and f : [r]n → {0, 1} be its indicator function. Denote α = µ(F) = E[f ]. In the next lemma we give some explicit constraints on the Fourier weight of f . 17
Lemma 4.3 Let f
: [r]n → {0, 1} be an indicator function of an independent set
of G. Write f in its Fourier decomposition: X
f=
fˆ(S)χS
S∈[r]n
Then for every j = 0, 1 . . . t − 1 holds X s≥j
Proof
1 − r−1
s s w(s) = 0 j
(4)
We write f in its Fourier decomposition and calculate f T r AG f . On one
hand it equals 0 as f is an indicator function of an independent set. One the other hand
0 = hf, AG f i =
* X
+ fˆ(S)χ(S),
S
X
λS fˆ(S)χ(S)
S
=
X
λS |fˆ(S)|2 =
X
λs w(s)
s≥0
S
Substituting λs from Proposition 4.2 we get ! s X t−1 X 1 j s j j − (−1) r X w(s) = 0 r−1 j j=0 s≥0 That is the sum is 0 as a polynomial in X . For each j the coe cient of X j equals s X s 1 j j (−1) r − w(s) r − 1 j s≥0 which equals 0 and the lemma follows. We are now ready to prove the st part of the theorem. We take f : [r]n → {0, 1} as above. We show that α = E[f ] ≤ 1/rt .
Proof of Theorem 4.1
For j = 0, 1 . . . t − 1 let βj = (−1)j ·
rt −rj r t −1
. We sum
the equations (4) over j 's with weights βj and get s t−1 X X 1 s 0 = βj − w(s) r − 1 j j=0 s≥0 s X ! t−1 X 1 s = − βj w(s) r−1 j j=0 s≥0 s X ! t−1 1 X 1 s = w(0) + t − (−1)j (rt − rj ) w(s) r − 1 s≥1 r−1 j j=0 X 1 · Q(s)w(s) = α2 + t r − 1 s≥1 18
where Q(s) denotes the function in the summation:
Q(s) =
1 − r−1
s
! s (−1)j (rt − rj ) j j=0
t−1 X
for s ∈ N
We show in the appendix (Claim A.1) that Q(s) ≥ −1 and attains its minimum only at s = 1 . . . t. Thus we get the following inequality: 1 X 1 0 ≥ α2 − t w(s) = 0 ≥ α2 − (α − α2 ) · t r − 1 s≥1 r −1
Upper Bound of the theorem
which implies the
α ≤ 1/rt The proof of
Uniqueness
and
Stability
are essentially reduced to proving
Theorem 3.1 in Section 3.
•
Uniqueness:
Assume µ(F) = α = 1/rt . Then
α2 − (α − α2 ) · and thus
P
−1 (since
Ps≥1
s≥1
X 1 1 2 = 0 = α + · · Q(s)w(s) rt − 1 rt − 1 s≥1
Q(s)w(s) = −(α − α2 ) That is w(s) 6= 0 implies Q(s) = w(s) = α − α2 ). Then, by Claim A.1, the weight of f is
concentrated on levels 0 through t its Fourier transform. Therefore we can apply Theorem 3.1 to conclude that f is a t-umvirate as required.
•
Stability:
Assume µ(F) = α ≥ 1/rt − for small enough > 0. Then
α2 − (α − α2 ) rt1−1 ≥ −C 0 for some C 0 = C 0 (r, t) > 0. Thus α2 − (α − α2 ) ·
X 1 1 0 2 + C ≥ 0 = α + · · Q(s)w(s) rt − 1 rt − 1 s≥1
and therefore
X
(Q(s) + 1)w(s) ≤ (rt − 1)C 0 = C 00
s≥1
Since Q(s) = −1 for s = 1 . . . t, and using the moreover part of Claim A.1, which says that Q(s) ≥ −1 + η(r, t) for s > t, we get X X η w(s) ≤ (Q(s) + 1)w(s) ≤ C 00 s>t
s>t
19
which implies that most of the Fourier weight of f is concentrated on levels P C 00 t 0 through t, i.e. s≥t w(s) ≤ η . Since E[f ] ≤ 1/r , we can apply the Theorem 3.1 to conclude that there is some t-umvirate g : [r]n → {0, 1} such C 00 C, η
that Prx [f (x) 6= g(x)] ≤
where C = C(r, t) from the Theorem 3.4. In
other words there is some t-umvirate family G ⊆ [r]n such that µ(G4F) ≤ K, where K = K(r, t). This completes the proof of the theorem.
Remark
Observe that in fact we had more information than actually required
for Theorem 3.1. In the uniqueness part if we assume that E[f ] = 1/rt , then solving the linear equations (4) gives us the Fourier weight of f on each level. One can easily check, not surprisingly, that the weight equals exactly to that of a
t-umvirate. Then given a function f : [r]n → {0, 1} with all this information, it is easy to derive that f is indeed a t-umvirate.
5
General graph products
In this section we try to generalize the techniques from the previous section, by applying it on more general graphs. Recall the de nition from Section 2. For a t
xed graph H = ([r], E), H ⊗n is a graph on vertices [r]n , where (x1 . . . xn ) and
(y1 . . . yn ) are not connected if (xi , yi ) ∈ / E for some t coordinates. Let A be the normalized adjacency matrix of H . Since A is symmetric the eigenvalues are real and the eigenvectors form an orthonormal basis of Rr . Let
1 = λ0 ≥ λ1 ≥ · · · ≥ λr−1 be the eigenvalues of A. Denote the smallest eigenvalue P of A by λ∞ . Note that λ∞ < 0 as λi = tr(A) and λ0 = 1. Let v0 = 1, v1 , . . . , vr−1 be the corresponding orthonormal eigenbasis. For each S ∈ [r]n we de ne
v S = v S1 ⊗ v S2 ⊗ · · · ⊗ v Sn Since {vi } is an orthonormal basis of Rr , the set {vS : S ∈ [r]n } forms an orn
thonormal basis of Rr and therefore any function f : [r]n → R can be written as
f=
X
fˆ(S)vS
where
S∈[r]n
20
fˆ(S) = hf, vS i
(5)
In particular, one can prove a the analogs of Proposition 2.5. Let f : [r]n → R. Then
E[f ] = fˆ(0)
and
E[|f |2 ] =
X
|fˆ(S)|2
S∈[r]n
In particular if f : [r]n → {0, 1} is a boolean function, then X X X |fˆ(S)|2 w(s) where w(s) = E[f ] = E[|f |2 ] = |fˆ(S)|2 = S∈[r]n
s≥0
S:|S|=s
5.1 Main Theorem for general base graphs Let H = ([r], E) be some xed graph with independence number α = α(H). We t
t
are interested in large independent set of H ⊗n . Recall that the vertices of H ⊗n are strings of length n over [r], and (x1 . . . xn ) and (y1 . . . yn ) are not connected in t
H ⊗n if (xi , yi ) is not connected in H for at least some t coordinates. Let I be an independent set of H of size |I| = αr. One type of large independent t
set of H ⊗n can be obtained by taking all strings (x1 . . . xn ) such that x1 , x2 . . . xt ∈
I . Another type can be obtained by taking strings such that at least t + 1 out of x1 , x2 . . . xt+2 are in I . More generally one can take all strings in which at least t+i of x1 , x2 . . . xt+2i are in I . Moreover, the independent sets in dierent coordinates might be dierent. That is for some I1 . . . It+2i independent sets of H and for some t
xed coordinates 1 ≤ j1 < · · · < jt+2i ≤ n, we de ne an independent set of H ⊗n to be the set all of strings (x1 . . . xn ) such that xjk ∈ Ik for at least t + i values of
k . We call such independent sets 't + i out of t + 2i' type of independent set. We want to prove an analogue of Theorem 4.1 of the following type: If H t
satis es certain properties, then α(H ⊗n ) = α(H)t . Moreover, any independent set
J of this size has the structure of t-umvirate, i.e. is obtained by xing t coordinates i1 . . . it and t independents sets I1 . . . It of size α(H) and letting J be the set of all strings (x1 . . . xn ) such that xi1 ∈ I1 . . . xit ∈ It . An easy computation shows that if α(H) ≥
1 t+1
then 't + 1
of t + 2' type of
independent set is at least as large as α . Therefore it is necessary to assume t
α(H) <
1 . t+1
Unfortunately this condition is not enough as shown in subsection
5.3. By making additional assumptions on the eigenvalues of H we are able to t
prove that α(H ⊗n ) = α(H)t . We also give some indication that large independent sets must have the structure of t-umvirate, however we are unable to show that.
21
Theorem 5.1 Let t ≥ 2 and H = ([r], E) be a connected non-bipartite graph with
α = α(H) the independence number of the graph. Let A ∈ Rr×r be the normalized
adjacency matrix of H , and denote its eigenvalues by 1 = λ0 > λ1 ≥ · · · ≥ λr−1 = t λ∞ . Let G = H ⊗n . Assume that H satis es the following conditions: • α=
−λ∞ 1−λ∞
• α<
1 t+1
•
λ1 −λ∞
(equivalently |λ∞ | < 1t )
n o 1 < min 1, (1−λ∞ t ) −1
Then α(G) = α(H)t .
Remarks • The rst assumption is the equality in Homan's bound, α ≤
−λ∞ . λ0 −λ∞
Since
we are relying on the spectral methods, it is natural to make this assumption here.
• We did not make assumptions about the regularity of H , and in general it can be thought of as a reversible Markov chain. Thus the theorem can be applied, for example, on regularizable graphs. These are the graphs that a regular graph can be obtained from them by replacing each edge by some positive number of edges. Berge [Ber78] characterized such graphs as those satisfying the condition that every non-empty independent set S ⊆ V has more neighbors than elements, i.e. |N (S)| > |S|. Of course, the eigenvalues should be of the regular matrix so that v0 = 1.
• It is clear that α(G) ≥ αt , e.g. by taking some independent set I of H of size αr, and considering (x1 . . . xn ) such that x1 , x2 . . . xt ∈ I . Therefore we need to show that α(G) ≤ αt .
• The third condition,
λ1 −λ∞
n o 1 < min 1, (1−λ∞ , is not very clear. It is simply )t −1
required by the technicalities of the proof. We do not know whether both inequalities in this condition are really necessary. Just like in the proof of Theorem 4.1 we de ne BG , a pseudo-adjacency matrix of
G. De ne an r × r matrix B over the ring R[X]/(X t = 0) to be B = (1 − rX)A + XJr 22
t
where Jr is the all ones matrix. One can easily check that BG = B ⊗n is a pseudoadjacency matrix of G. The eigenvalues of BG can be related to the eigenvalues of
A:
Claim 5.2 For S ∈ [r]n let λS =
Qn
i=1
λSi Then
1. The eigenvalues of B are λB = λ0 0
for j = 1 . . . r − 1
λB = λj (1 − rX) j
and the corresponding eigenvectors are vj . t
2. The eigenvalues of BG = B ⊗n are µS = λ S
t−1 X
(−r)i
i=0
! |S| Xi i
where λS =
Y
λSi for S ∈ [r]n
and the corresponding eigenvectors are vS = vS1 ⊗ · · · ⊗ vSn .
Proof 1. The eigenvalues of B are straightforward computation. Let λj be the j 'th eigenvalue of A and vj be the corresponding eigenvector. If j = 0, then
vj = 1 and B1 = (1 − rX)1 + rX1 = 1 and for vi ⊥ 1 we have
Bvi = (1 − rX)Avi + XJvi = (1 − rX)λi vi Item 1 follows by recalling that λ0 = 1. 2. One can easily check that the eigenvectors of BG are vS = vS1 ⊗ · · · ⊗ vSn and the corresponding eigenvalues are
µS =
n Y j=1
λB Sj =
Y
λSj (1 − rX) = λS (1 − rX)|S|
j:Sj 6=0
Since X t = 0, we need to compute the coe cients of X j for j = 0 . . . t − 1. Easy calculations show that it equals (−r)j |S| λS . j 23
Now if f : [r]n → {0, 1} is an indicator function of some independent set of G of size α(G), we can write it as in equation (5) X f= fˆ(S)vS S∈[r]n
where fˆ(0) = E[f ] = α(G). Since f indicates an independent set of G, analogously to Lemma 4.3, we get the following equality * + t−1 X X X X X |S| 2 i 0 = hf, Gf i = fˆ(S)vS , fˆ(T )µT vT = µS |fˆ(S)| = (−r) λS |fˆ(S)|2 X i i i=0 S T S S This equality can be understood as an equality of polynomials, implying X |S| λS |fˆ(S)|2 = 0 for i = 0 . . . t − 1 i n
(6)
S∈[r]
Now we are ready to prove the main theorem of this section.
Proof of Theorem 5.1 we let βj = (−1) · j
γ t −γ j γ t −1
Denote 1 −
1 λ∞
by γ . Note that γ =
1 α
> t + 1. Now
for j = 0, 1 . . . t − 1. We sum the equations (6) over j 's
with weights βj and get t−1 X
X |S| 0 = βj λS |fˆ(S)|2 i j=0 S∈[r]n ! t−1 X X |S| λS βj |fˆ(S)|2 = i n j=0 S∈[r]
t−1 1 X X |S| ˆ j t j = w(0) + t λS (−1) (γ − γ ) |f (S)|2 γ − 1 S6=0 i j=0 X 1 = α(G)2 + t Q(S) · |fˆ(S)|2 γ − 1 S6=0 where Q(S) : [r]n → R denotes the function in the sum: t−1 X |S| j t j Q(S) = λS (−1) (γ − γ ) i j=0 We show in the appendix (Claim A.2) that Q(S) ≥ −1 and attains this minimum only at S that satisfy |S| = 1 . . . t and λS = (λ∞ )|S| . Here we use the third condition 24
of the theorem. This is exactly the condition that assures that Q(S) ≥ −1 and attains its minimum on the correct values of S . Thus we get 1 1 X ˆ |f (S)|2 = α(G)2 − (α(G) − α(G)2 ) t 0 ≥ α(G)2 − t γ −1 γ −1 S6=0
which implies that
α(G) ≤ 1/γ t = αt as required.
Remarks • (Compare to the remark in the end of the previous section) In fact we proved a little more. Let f : [r]n → {0, 1} be an indicator of J , an independent set of
G of maximal size, µ(J) = αt . Then all its Fourier weight is concentrated on levels 0 through t. Moreover f is spanned by (n-tensor of) the eigenspace of
λ∞ together with 1. In addition, the equations (6) give us t linear equations in t variables: w(1) . . . w(t). The solution is exactly the weight of any function
g : [r]n → {0, 1}, that is an indicator of a t dimensional cube G with each side of the correct size, e.g. G = [αr]t × [r]n−t . It is appealing to try and prove the analogue of Theorem 3.1 with the additional information of weights on each of the t levels. While Theorem 3.1 required only the expectation of f , even given the weight of all t levels doesn't assure that the function has the required structure. We give a speci c counter example for r = 27, t = 2, αr = 6 (note that α <
1 ). t+1
We describe a set G ⊆ [27]3 that depends on all 3 coordinates but its Fourier weight at each level equals that of the set [6] × [6] × [r], that is E[G] =
wG (1) =
112 812
and wG (2) =
G=
196 . 812
G is for example
{1 . . . 9} × {1, 2, 3} × [r]
4 , 81
×
{4}
[r]
!
× {1 . . . 9}
We skip the computation of the weights of G .
• If the maximal independent sets indeed have the structure of t-umvirate, then, we believe, the stability of this result should follow analogously to the proof of Stability in 3.1.
25
5.2 Examples of graphs that satisfy the conditions of Theorem 5.1 Tensor graphs:
The simplest example is H = Kr for appropriate values of r
and t. Clearly the theorem can also be applied on tensor product of two rcliques. More generally, if two graphs H1 and H2 satisfy the assumptions of the theorem, consider H = H1 ⊗ H2 . Let µ1 , µ∞ be the relevant eigenvalues of H1 and ν1 , ν∞ be the relevant eigenvalues of H2 . For the third condition to hold we need e.g. |µ∞ | ≥ |ν∞ | and also µ1 ≥ ν1 . Then λ1 (H) = max {µ1 , µ∞ ν∞ } and λ∞ (H) = µ∞ . An independent set of size α(H1 ) can be obtained by taking all pair (v, w) s.t. v is in some maximum independent set of H1 and w in H2 .
Kneser graph:
Let k, d be two positive integers, and let m = 2k + d. The Kneser
graph KG(m, k) is the graph whose vertices are k -element subsets of a set of m elements, where two vertices are connected if and only if the two corresponding sets are disjoint. The eigenvalues of Kneser graphs are well known (see e.g. (k+d−j ) d [Lov79]), and they are of the form (−1)j k+d for j = 0, 1 . . . k . In particular ( d ) k(k−1) k −λ∞ λ∞ = − k+d and λ1 = (k+d)(k+d−1) . It is also well known that α = k/m = 1−λ ∞ (this is known as the EKR theorem [EKR61]). We have α <
1 t+1
if and only if d > k(t − 1). One can check that for such
parameters the third condition of the theorem is also satis ed and thus it can by applied for d > k(t − 1). Next we give two examples of families of strongly regular graphs. A k -regular graph
H on v vertices is said to be strongly regular with parameters λ and µ, denoted by srg(v, k, λ, µ), if every two adjacent vertices have λ common neighbors and every two non-adjacent vertices have µ common neighbors. It is well known that the eigenvalues of an srg(v, k, λ, µ) are λ =
√ (λ−µ)± ν , 2k
where ν = (λ − µ)2 − 4(k − µ)
For more information see the excellent book by Chris Godsil and Gordon Royle [GR01].
Paley graph:
Let a be an even integer and let q = pa be a prime power so that
q ≡ 1 mod 4. Note that this implies that the unique nite eld of order q , Fq , has a square root of -1. Now let V = Fq and (a, b) ∈ E if and only if (a − b) is a square in Fq . This is well de ned since a − b = −1(b − a), and 26
since −1 is a square, it follows that a − b is a square if and only if b − a is a square. A Paley graph Pq is a strongly regular graph srg(q, q−1 , q−5 , q−1 ). It has two 2 4 4 eigenvalues beside λ0 = 1: λ1 =
1√ 1+ q
1 and λ∞ = − √q−1 . For q = pa is a
−λ∞ square, the independence number of Pq is α(Pq ) = √1q = 1−λ (see [BDR88]). ∞ √ 1 We have α < t+1 if and only if q > t + 1. One can check that for these
parameters the third condition of the theorem is also satis ed and thus it can by applied for q > (t + 1)2 .
Orthogonal arrays graph:
Another type of strongly regular graph can be ob-
tained from orthogonal arrays. A q 2 × k array over [q] is OA(q, k) if every two columns of A contain each possible pair of elements of [q] exactly once. For more see [HSS99] We de ne a graph G from A as following: The vertices of G are the rows of the A, and two vertices are connected if the corresponding rows agree in some coordinate. It is easy to check that G is an srg(q 2 , (q − 1)k, q − 2 + (k − q−k 1 and λ∞ = − (q−1) . For the k(q−1) n o third condition to hold we require k > max 2q , q(1 − ( q−1 )t ) . If the graph q
1)(k − 2), k(k − 3)). Its eigenvalues are λ1 =
contains an independent set of size q , then we are in the setup of the theorem. The complement graph G is also a strongly regular graph srg(q 2 , (q − k +
1)(q − 1), (q − k)2 + k − 2, (q − k)(q − k + 1)). Its eigenvalues are µ1 = 1 and µ∞ = − (q−1) . For the third condition to hold we require o k < min 2q + 1, q( q−1 )t + 1 . It contains an independent set of size q , e.g. q k−1 (q−k+1)(q−1) n
all rows that have 1 in the rst coordinate. To see a concrete example let q be a prime power and let F = GF (q). Pick arbitrarily k distinct elements x1 . . . xk ∈ F. We de ne A to be a q 2 ×k matrix with rows indexed by pairs of (a, b) ∈ F2 . Let A((a, b), j) = axj + b. It is easy to see that A is indeed an orthogonal array and the corresponding graph contains an independent set of size q , e.g. for some xed a0 ∈ F take all the rows indexed by {(a0 , b) : b ∈ F}. Thus for appropriate values of q and k we get a strongly regular graph (or its complement) that satis es the conditions of the theorem.
27
5.3 A counter-example In this section we show a graph G with α(G) <
1 , t+1
t
but α(G⊗n ) > α(G)t . In order
to do that, we rst prove the following lemmas.
Lemma 5.3 Let H = (V, E) and denote the minimal degree of H by δ. Then for
every > 0 there is n s.t. α(H ⊗n ) ≥
1 1+δ
− .
That is if H is not regular, then α(H ⊗n ) can be bounded as a function of the minimal degree. This lemma can look surprising at rst, as the naive bound is
α(H) >
Proof
1 , 1+∆
where ∆ is the maximal degree of H .
Take v ∈ V s.t. deg(v) = δ and let M = V \ (N (v) ∪ {v}), where N (v)
is the neighborhood of v . Consider the following (disjoint) subsets of V n :
Ij = M j−1 × {v} × V n−j
1≤j≤n
Clearly Ij are independent sets as any vertex in Ij has v in its j 'th coordinate. De ne
I = ∪nj=1 Ij This set is also an independent set. Let x ∈ Ii and y ∈ Ij , where i < j . Then
xi = v and yi ∈ M and thus (xi , yi ) ∈ / E . Since Ij are disjoint we get n X 1 |M | |M |n−1 1 |M | n µ(I) = + + ··· + = 1−( ) µ(Ij ) = 2 n |V | |V | |V | δ + 1 |V | j=1 And the lemma follows by taking n such that
|V |−δ−1 |V |
n
< .
This lemma is not applicable to d-regular graphs, as it is no better than the naive bound of α(H ⊗n ) ≥
1 . 1+d
So instead of taking a vertex of minimal degree,
we may take an independent set in H and consider its neighborhood. This is done in the following lemma.
Lemma 5.4 Let H
= (V, E) contain an independent set J ⊆ V s.t. |J| = a and
|N (J)| = b. Then for every > 0 there is n s.t. α(H ⊗n ) ≥
Proof
a a+b
− .
The proof is very similar to the previous, by letting M be V \ (J ∪ N (J))
and de ning Ij = M j−1 × J × V n−j for 1 ≤ j ≤ n. Then I = ∪nj=1 Ij is an |M | n a ⊗n independent set in H and µ(I) = a+b 1 − ( |V | ) , which proves the lemma. 28
t
Now we would like to prove analogous lemma for H ⊗ n . We rely on a simple t
observation that any independent set of (H ⊗n )⊗ t is also an independent set of t
H ⊗ tn , where we think of vertices of the former as strings of length tn divided into t parts each of length n. We get the following result.
Lemma 5.5 Suppose H = (V, E) contains an independent set J ⊆ V s.t. and |N (J)| = b. Then for every > 0 there is n s.t. α(H
Proof
⊗t n
)≥
By the previous lemma there is n such that α(H ⊗n ) ≥
a t a+b a a+b
|J| = a
− .
− . Let I be t
an independent set of H ⊗n of this size. We de ne an independent set of H ⊗ tn to be all x1 . . . xtn such that xjn+1 . . . x(j+1)n ∈ I for all j = 1 . . . t. The independent a set is of size ( a+b − )t and the claim follows. t
We now describe a regular graph H with small α(H), but α(H ⊗n ) > α(H)t . It is a variation of the example appearing in [ADFS04]. 1. Take the complete 2k -partite graph, each part of size d. The vertices are
{vi,j : 1 ≤ i ≤ 2k, 1 ≤ j ≤ d}, and there is an edge between vi,j and vi0 ,j 0 i i 6= i0 . 2. Remove the edges hv2i−1,d , v2i,d i for i = 1 . . . k . 3. Take a clique on (2k −1)d+1 vertices. Denote the vertices by u1 . . . u(2k−1)d+1 . 4. Remove the edges hu2i−1 , u2i i for i = 1 . . . k . 5. Add edges hvi,d , ui i for i = 1 . . . 2k . The graph is (2k − 1)d regular on r = (4k − 1)d + 1 vertices. The independence number of H is α(H) =
d+2 r
≈
1 4k−1
(for large d). Now let J be an independent set of
some d-part, e.g. J = {v1,j : 1 ≤ j ≤ d}. Then N (J) is the rest of the vi,j 's together |J| |J|+|N (J)|
with u1 . Thus |N (J)| = (2k − 1)d + 1. Therefore t
α(H). Therefore by Lemma 5.5 we get that α(H ⊗n ) ≈
6
= 1 t
2k
d 2kd+1
1 1 ≈ 2k > 4k−1 ≈ t 1 > 4k−1 = α(H).
Concluding Remarks and Future Directions
We mention several open questions.
29
• Following the Theorem 1.2 of Ahlswede and Khachatrian [AK98], recall the t-intersecting families from (1): n−t 2 n The theorem says that the only maximal t-intersecting family of [r] are isoFi = {x ∈ [r]n : at least t + i of x1 . . . xt+2i equal 0}
0≤i≤
morphic to Fi for the appropriate values of i. It is natural to conjecture that
Fi 's are not only maximal but also stable.
Conjecture 6.1 Fix any natural r ≥ 3 and t ≥ 2. An let µmax be the size of maximal t-intersecting family in [r]n , that is
M = max {µ(F ) : F − t-intersecting family} = max {µ(Fi )} i
Then, not only all the maximal t-intersecting family of [r]n are isomorphic to some Fi , but also these families are stable. That is there is K = K(r, t) s.t. the following holds: ◦ Assume (r − 2) does not divide (t − 1). Then for any > 0 and any t-intersecting family G of size at least M − holds µ(G 4 F) < K t−1 . for some F isomorphic to Fb r−2 c ◦ Assume (r − 2) divides (t − 1). Then for any > 0 and any t-intersecting family G of size at least M − holds
µ(G 4 F) < K t−1 t−1 . for some F isomorphic either to F r−2 −1 or to F r−2 t
• It is easy to see that for every t the function α(G⊗n ) is monotone increasing with n. Since it is bounded by above by 1, we conclude that there is a limit. Denote it by αt (G). t
t
αt (G) := lim α(G⊗n ) = sup α(G⊗n ) n
n
t
We also know that α(G⊗n ) ≥ α(G⊗tn )t and thus
αt (G) ≥ (α1 (G))t This leads to the following question. 30
1 Question 6.2 Is there a graph G for which α1 (G) ≤ t+1 and αt (G) > (α1 (G))t ?
The condition α1 (G) ≤
1 t+1
implies that an independent set of type 't + i out
of t + 2i' is smaller than (α1 (G))t for every i ≥ 1. An analogous question can be asked for any α1 (G) <
1 2
as well (note that if α1 (G) >
1 2
then α1 (G) = 1).
• It is shown in [ADFS04], using a Katona-type proof, that for a given vertextransitive graph G holds α1 (G) = α(G). One could strengthen the question above for vertex transitive graphs:
Question 6.3 Is there a vertex-transitive graph α(G) ≤
1 t+1
and αt (G) > (α(G)) ?
G which satis es α1 (G) =
t
• Another question related to the example in Section 5.3 is the question of nding α1 (G) for a given graph. We showed that
α1 (G) ≥ max I
|I| |I| + |N (I)|
where the maximum is taken over all independent sets of G and N (I) denotes the neighborhood of I . It is not clear whether this is the correct bound of
α1 (G), or there are graphs that overcome it. The quantity α1 has been introduced in [BNR96], where it is called the Ultimate Categorical Independence Ratio of G. For related discussion see [AL07].
• The following question can be thought of as a generalization of Theorem 3.1. Suppose we are given a set F ⊆ [r]n , and let f : [r]n → {0, 1} be its indicator function. Assume all the Fourier weight of f is concentrated on levels 0 through t. Can we say that F can be decomposed into a disjoint union of
t-umvirates? The answer to this question is 'No'. We give a counter-example by showing F ⊆ [3]3 whose Fourier weight is concentrated up to the second level:
F=
! {23} × {23} × {23} {1}
×
{1}
×
{1}
That is F contains all strings over {2, 3} together with the string 111. In order to see that F is indeed concentrated up to the second level, observe that the complement of F can be decomposed into a disjoint union of
31
duumvirates:
{1}
×
{23}
× {123}
F c = {123} ×
{1}
×
{23}
× {123} ×
{23} {1}
Since the duumvirates are disjoint, the sum of their indicating functions is the indicating function of F c . And since the Fourier weight of each duumvirate is concentrated on levels 0 through 2, the Fourier weight of F c is also concentrated on these levels. We ask the following question about sets F ⊆ [r]n whose Fourier weight is concentrated on low levels.
Question 6.4 For a given set
F ⊆ [r]n assume that all its Fourier weight
is concentrated on levels 0 through t. Is it true that either F or F c can be decomposed into a disjoint union of t-umvirates?
A positive answer to this question would imply that any F ⊆ [r]n of small size, whose Fourier weight is concentrated on low levels, depends on a small number of coordinates. Together with the approach similar to the proof of Stability of Theorem 3.1 we could re ne the theorem of Kindler and Safra and make the size of junta dependent on the size of F .
• We conclude this section with a question related to the previous one. This question is of a combinatorial avor. Given a set F ⊆ [r]n that is a union of
k disjoint t-umvirates, what is the maximal possible number of coordinates that F may depend on? For example for 1 ≤ k ≤ r it is easy to see that F may depend on at most k(r − 1) + 1 coordinates.
32
Part II
On the Conditional Hardness of Coloring a 4-colorable Graph with Super-Constant Number of Colors
33
Abstract
For 3 ≤ q < Q we consider the
ApproxColoring(q, Q) problem of deciding whether
χ(G) ≤ q or χ(G) ≥ Q for a given graph G. Hardness of this problem was shown in [DMR06]
for q = 3, 4 and arbitrary large constant Q under certain assumption regarding the hardness of Label Cover problem. We extend this result to values of Q that depend on the size of a given graph. The extension depends on the parameters of the conjectures we consider. In particular we show that under some assumptions it is NP-hard to color a 4-colorable graph with lgc (n) colors for some constant c > 0.
34
7
Introduction
Graph Coloring is one of the most fundamental problems in combinatorics and computer science. A graph G on n vertices is said to be q -colorable if there is an assignment from {1, . . . , q} to every vertex so that every two neighboring vertices receive dierent colors. The chromatic number of G, denoted by χ(G), is the minimal number q such that G is q -colorable. For q < Q we consider the
ApproxColoring(q, Q): Given a graph G, decide whether χ(G) ≤ q or χ(G) ≥ Q. It is well known that for any constant q ≥ 3 the problem ApproxColoring(q, q + 1) is NP-hard [Kar72]. Due to the self reducibility of the coloring problem, ApproxColoring(q, Q) problem
is computationally equivalent to coloring a q -colorable graph with Q − 1 colors. If we consider q to be some small xed number (e.g. 3 or 4), there is a huge gap between the values of Q for which an e cient algorithm is known, and that for which hardness results exist. For example for q = 3 the best known polynomial time algorithm is due to Chlamtac [Chl07]. The semi-de nite programming based algorithm solves the problem for Q = O(n0.2072 ) colors (see also [AC06], [BK97], [KMS98]). It is interesting to note that currently known techniques seem to have limitations of Q = nΩ(1) (see e.g. [FLS04]). On the other hand the strongest known hardness result shows that the problem is NP-hard for Q = 5 (see [GK05], [KLS00]). So the problem is open for all 5 < Q < O(n0.2072 ). Most of the inapproximability results are shown by a reduction from the Label Cover problem. In the Label Cover problem, we are given a bipartite graph G =
(V ∪ W, E), a number R and a set of constraints Π = {πe : e ∈ E}. Each edge e ∈ E is associated with a subset of πe ⊆ {1, . . . , R} × {1, . . . , R}. We say that a constraint πvw is satis ed by an assignment L : V ∪ W → {1, . . . , R} if
(L(v), L(w)) ∈ πvw . The goal is to nd an assignment that maximizes the fraction of satis ed constraints. The value of G, denoted by val(G), is the fraction of satis ed constraints under such assignment. It is convenient to think of the Label Cover problem as a "2-prover 1-round game" in which the veri er picks at random two queries (corresponding to an edge (v, w) ∈ E ) according to some distribution, and sends one query to each prover (v to the rst prover and w to the second) The provers' goal is to convince the veri er to accept. The value of the game is the maximum, over all provers' strategies, of the acceptance probability of the veri er. 35
We say that a Label Cover instance has a projection property if the answer of the rst prover uniquely determines the answer of the second prover. In other words, the constraints are projections πe : RV → RW . Using the PCP theorem [ALM+ 98, AS98] together with the Raz's Parallel Repetition Theorem [Raz98] it is known that for any > 0 and R ≥ poly( 1 ) it is NP-hard to determine whether the value of given instance of Label Cover with the projection property is 1 or it is at most . However, such version of Label Cover seems not to be strong enough to obtain hardness result of some NP-hard problems like Vertex Cover and Graph Coloring. A stronger version called "unique 2-prover games" can be considered. In the unique games the answer of the second prover uniquely determines the answer of the rst prover and vice versa, that is the constraints can be thought of as permutations of
{1, . . . , R}. Khot's unique games conjecture [Kho02] is the following:
Conjecture 7.1 (Unique Games Conjecture) For any constants δ > 0, there is some R = R(δ) such that it is NP-hard to determine whether a unique 2-prover game with answers from a domain of size R has value at least 1 − δ or at most δ .
The shortcoming of this kind of conjecture is that one can determine in polynomial time whether a unique 2-prover game has value 1, i.e. it is clearly not NP-hard to check for perfect completeness. To overcome this Khot [Kho02] also introduced d-to-1 games. We say that a 2-prover game has "d-to-1 property" if the answer of the rst prover uniquely determines the answer of the second prover and for every answer of the second prover, there are at most d answers for the rst prover for which the veri er would accept. A conjecture analogous to the above can be considered for d-to-1 games [Kho02]:
Conjecture 7.2 (d-to-1 Conjecture) For any > 0 there is R = R() s.t. the
following problem is NP-hard. Given a bipartite d-to-1 Label Cover instance Φ = (V ∪ W, E) with label sets {1, . . . , R} for V and {1, . . . , dR} for W , distinguish between the case where val(Φ) = 1 and the case where val(Φ) < . Assuming Khot's 2-to-1 conjecture it is shown in [DMR06] that the problem of coloring a 4-colorable graph with any constant number of colors is NP-hard. We give here a quantitative version of this result. Speci cally we analyze the dependency between the soundness of the Label Cover problem and the hardness 36
of nding a legal coloring of a graph with small chromatic number. The main result is the following theorem:
Theorem 7.3 Assume that given a bipartite 2-to-1 Label Cover instance
Φ with
the label set of size R = O(lg(n)) it is NP-hard to distinguish between the following cases: • val(Φ) = 1 • val(Φ) <
1 f (n)
for some f (n). Then it is NP-hard to color a 4-colorable graph with f c (n) colors for some constant c > 0. For example if f (n) = lgδ (n) then it is NP-hard to color a 4colorable graph with lgcδ (n). The theorem improves the dependency between the soundness of 2-to-1 La1 bel Cover ( f (n) ) and the hardness of the coloring problem. For comparison, the
(implicit) dependency in [DMR06] is logarithmic, i.e. the soundness of 1/f (n) in the Label Cover is translated into hardness of coloring a 4-colorable graph with
Ω(log(f (n))) colors. Two main technical results used are called the ation of the
invariance principle and a vari-
Majority is Stablest theorem, which have been developed in the paper
of Mossel et al. [MOO05] and generalized in the paper of Dinur et al. [DMR06]. In section 9 we prove a variant of the Majority is Stablest theorem following [DMR06] with adjustments for our purposes. Then, in Section 10, we present the reduction of [DMR06] from 2-to-1 Label Cover problem to the Graph Coloring problem and work out the parameters in the soundness of the reduction.
8
Preliminaries
8.1 Functions on the q-ary hypercube Let q be a xed integer. Let [q] denote the set {0, . . . , q − 1}. For an element
x ∈ [q]n denote by |x| the number of nonzero coordinates of x. Consider the space of real valued function with domain [q] or, equivalently, a vector space Rq with
37
inner product de ned as q
1X hv, wi = E[vw] = vi wi q i=1 and norm of a vector de ned as
kvk =
p
hv, vi
Let α0 = 1, α1 , . . . , αq−1 be some orthonormal basis of Rq . It de nes naturally an n
orthonormal basis of Rq by applying the n-fold tensor product. It is easy to see n
that the set {αx = αx1 ⊗ αx2 ⊗ · · · ⊗ αxn ∈ Rq : x ∈ [q]n } is indeed an orthonormal n
basis of Rq . Equivalently, we can think of αx as a function from [q]n to R de ned Q by αx (y) = ni=1 αxi (yi ). Thus any function f : [q]n → R can be written as X fˆ(αx )αx (7) f= x∈[q]n
It immediately follows by orthonormality that sX kf k = fˆ(αx )2 x∈[q]n
Next we de ne the notion of in uence of a variable on a function, introduced to computer science by Ben-Or and Linial in [BOL89].
De nition 8.1 Let f : [q]n → R be a function on a q-ary hypercube. The in uence of the i'th variable on f , is de ned as
Inf i (f ) = Ex\i [Varxi [f (x)|x1 , . . . , xi−1 , xi+1 , . . . , xn ]]
where x1 , . . . , xn are uniformly distributed in [q]. Note that for f : {0, 1}n → {−1, 1}, it holds that Inf i (f ) = Prx [f (x) 6= f (x1 , . . . , 1−
xi , . . . , xn )]. Some standard formulas are easily checkable using independence and orthonormality
Proposition 8.2 Let f : [q]n → R be as in (7). Then E[f ] = fˆ(α0 )
E[f 2 ] =
X
fˆ(αx )2
x
Var[f ] =
X
fˆ(αx )2
Inf i (f ) =
X x:xi 6=0
|x|>0
38
fˆ(αx )2
Analogously we can de ne "low-degree in uence", a notion useful in PCPs due to the fact that a bounded function cannot have too many coordinates with nonnegligible low-degree in uences.
De nition 8.3 The d-low-degree in uence of f : [q]n → R is X
Inf ≤d i (f ) =
fˆ(αx )2
x:xi 6=0,|x|≤d
The remark above follows from the following easy proposition
Proposition 8.4 Let f : [q]n → R be as in (7). Then X
Inf ≤d i (f ) ≤ d · Var[f ]
i
In particular for f : [q]n → [−1, 1] holds X
Inf ≤d i (f ) ≤ d
i
and thus there are at most d/ variables i with Inf ≤d i (f ) ≥ . Instead of picking x at random, changing one coordinate, and seeing how it changes the value of f , we can change a constant fraction (in expectation) of the coordinates.
De nition 8.5 Let f : [q]n → R, and let ρ ∈ [0, 1]. Suppose the string x is picked
uniformly at random and each coordinate yi is independently chosen to be xi with probability ρ and is a uniformly random element of [q] otherwise. We de ne the noise stability of f to be Sρ (f ) = E[f (x)f (y)]
Analogously we generalize the notion of stability with respect to two functions: Sρ (f, g) = E[f (x)g(y)] The notion above can be also considered as following: For any ρ ∈ [0, 1] de ne the following Markov operator on [q] (called the Bonami-Beckner operator) 1−ρ 1−ρ ρ + 1−ρ . . . q q . q .. .. . Tρ = . . .. .. 1−ρ 1−ρ 1−ρ . . . ρ + q q q 39
Clearly Tρ 1 = 1 and Tρ v = ρ·v for any vector v ⊥ 1. In particular holds T1 (f ) = f and T0 (f ) = E[f ]. The following formulas are standard and easily checkable
Proposition 8.6 Let f, g : [q]n → R be as in (7) with respect to some orthonormal basis {αi }. Then
Tρ⊗n (f ) =
X
ρ|x| fˆ(αx )αx
x
and by orthonormality
X |x| Sρ (f, g) = f, Tρ⊗n g = ρ fˆ(αx )ˆ g (αx ) x
By applying Tρ on a function f : [q]n → [0, 1] we get that the weight on higher P levels reduces exponentially. More precisely if g = Tρ f , then ˆ(αx )2 ≤ x:|x|≥k g P ρ2k fˆ(αx )2 ≤ ρ2k . We may think of Tρ as a smoothing operator. x
De nition 8.7 Let g : [q]n → R, and let η ∈ (0, 1). We say that g is η-smooth if P
x:|x|≥k
gˆ(αx )2 ≤ η k for all k ≥ 0.
8.2 Functions in Gaussian space Before we continue, we need to de ne some basic notions in L2 (Rn , γ), the space of real valued functions with domain Rn equipped with the standard Gaussian measure. The density function of the standard normal distribution is denoted by
γ(x) =
1 (2π)n/2
2
exp(− kxk ). The inner product is de ned as 2 Z hf, gi = Eγ [f g] = f (x)g(x)γ(x)dx Rn
For ρ ∈ [−1, 1] we denote by Uρ then Ornstein-Uhlenbeck operator p (Uρ f )(x) = Ey∼γ [f (ρx + 1 − ρ2 y)] For µ ∈ (0, 1) we de ne an indicator of half space function L2 (R, γ) as
Fµ (x) = 1x<Φ−1 (µ) (x) where Φ(t) =
Rt −∞
γ(x)dx is the cumulative distribution function.
A useful quantity that will appear later is hF , Uρ (1 − F1− )i for 0 < ρ < 1. Using the fact that
(1 − F1− )(x) = F (−x) 40
we get that Uρ (1−F1− ) = U−ρ F and therefore hF , Uρ (1 − F1− )i = hF , U−ρ (F )i. Observe that hF , U−ρ F i = Pr[X < Φ−1 (), Y < Φ−1 ()], where X and Y are −ρ correlated normal random variables with mean 0, variance 1. That is X ∼ N (0, 1) p and for Z ∼ N (0, 1) the r.v. Y is −ρX + 1 − ρ2 Z . It is shown in [RR01] that as → 0
hF , U−ρ F i ∼ 2/(1−ρ) (4π ln(1/))ρ/(1−ρ)
(1 − ρ)3/2 (1 + ρ)1/2
In particular if ρ < 1, then
hF , Uρ (1 − F1− )iγ = poly()
(8)
We want to de ne an analogue of f : [q]n → R in space of real valued functions with continuous domain. In order to do it we de ne rst a set of orthonormal functions Γx that correspond to αx .
De nition 8.8 (Continuous analogue of a function) Let
function with decomposition
f=
X
f : [q]n → R be a
fˆ(αx )αx
1 n Consider the (q − 1)n variables z11 , . . . , zq−1 , . . . , z1n , . . . , zq−1 We de ne Γx (z) = Qn i n i=1,xi 6=0 zxi . It is easy to see that the set {Γx : x ∈ [q] } forms an orthonormal set of functions. We de ne the continuous analogue of f to be the function fe : Rn(q−1) → R given by X
fe =
fˆ(αx )Γx .
The notion of in uence of a variable on a function fe is de ned analogously to the discrete case, Inf i (fe) = Ez1 ,..,zi−1 ,zi+1 ,..,zn [Varzi [f (z)|z1 , . . . , zi−1 , zi+1 , . . . , zn ]], i where zi stands for z1i , . . . , zq−1 independent normal random variables. Just like in P the discrete case we have Inf i (fe) = x:xi 6=0 fˆ(αx )2 . This holds by independence
for any orthonormal basis and and is not speci c to αx or Γx .
Proposition 8.9 Let
T be a symmetric linear operator on Rq with eigenvectors
1 = λ0 > λ1 ≥ · · · ≥ λq−1 > −1 and the corresponding orthonormal eigenbasis α0 = 1, α1 , . . . , αq−1 . De ne T˜ = Uλ1 ⊗ Uλ2 ⊗ · · · ⊗ Uλq−1 . Then for all x ∈ [q]n ! Y T˜⊗n Γx = λi Γx i:xi 6=0
41
Thus, by orthonormality of Γx , for any two functions f, g : [q]n → R as in (7) w.r.t. the basis {αi } holds
E D f, T ⊗n g = fe, T˜⊗n ge
8.3 The Majority is Stablest Theorem The Majority is Stablest Theorem [MOO05] roughly says that for all functions
f : [q]n → [0, 1] in which each coordinate has o(1) in uence, the noise stability of f is bounded by some function of E[f ]. More speci cally
Theorem 8.10 ([MOO05, Theorem 4.4]) Fix q ≥ 2 and ρ ∈ [0, 1]. Then for any > 0 there is a small enough δ = δ(, ρ, q) such that for any function f : [q]n → [0, 1] such that Inf i (f ) ≤ δ
∀i ∈ {1, . . . , n}
holds
Sρ (f ) ≤ FE[f ] , Uρ FE[f ] γ + In particular in case of q = 2 and a balanced functions f : {0, 1} → {0, 1} the theorem states that if
1 1 arcsin ρ + = Sρ (M aj) + Sρ (f ) > FE[f ] , Uρ FE[f ] γ + = + 4 2π then f has some in uential coordinate. That is among all boolean functions in which each coordinate has o(1) in uence, the
Majority function has the largest
noise stability. This theorem is generalized in [DMR06] in two directions: the stability is de ned with respect to two functions and for any Markov operator T on [q] (not only for
Tρ ). The idea is that given a symmetric Markov operator T with eigenvalues 1 = λ0 > λ1 ≥ · · · ≥ λq−1 , it is enough to bound its spectral radius ρ = r(T ) = max {|λ1 |, |λq−1 |} below 1. Suppose we are given a symmetric Markov operator T on [q] with spectral radius ρ < 1, and two functions f, g : [q]n → [0, 1] that satisfy the inequality
f, T ⊗n g > FE[f ] , Uρ FE[g] γ +
The main technical result in [DMR06, Theorem 3.1] says that in such case f and g have a common coordinate with non-negligible in uence. In our setup, however, we consider functions f and g with small expectation and ρ some xed constant and 42
thus we allow ourselves to consider the case of hf, T ⊗n gi > FE[f ] , Uρ0 FE[g] γ + , for some ρ0 > ρ. In return we conclude that f and g have a common coordinate with relatively large in uence on both functions. The exact formulation and the proof appear in the next section.
9
A Variant of the Majority is Stablest Theorem
In this section we prove a technical theorem which will be useful in the soundness of the reduction in Section 10. This is an analogue of Theorem 3.1 of [DMR06]. We adjust the proof for our purposes. The main change is in Lemma 9.7 (analogue of Lemma 3.9 in [DMR06]). Let q be a xed integer, and let T be a symmetric Markov operator on [q] with eigenvalues 1 = λ0 > λ1 ≥ · · · ≥ λq−1 > −1. Assume that ρ = r(T ) =
max {|λ1 |, |λq−1 |} < 1. Let α0 = 1, α1 , . . . , αq−1 be the corresponding eigenvectors. Now suppose that we are given two functions f, g : [q]n → [0, 1] that do not have a common in uential coordinate. We show that following bound on the quantity
hf, T ⊗n gi:
Theorem 9.1 Let q be a xed integer, and let T be a symmetric Markov operator
on [q] such that ρ = r(T ) < 1 and let ρ0 ∈ (ρ, 1). Then for any > 0 there are δ = O(1) and k = O(lg( 1 )), where the constants in the O notation depend only on ρ and ρ0 , such that the following holds: If f, g : [q]n → [0, 1] are two functions satisfying ≤k min Inf ≤k i (f ), Inf i (g) < δ
∀i
then
f, T ⊗n g ≥ hFµ , Uρ0 (1 − F1−ν )iγ −
(9)
f, T ⊗n g ≤ hFµ , Uρ0 Fν iγ +
(10)
and
where µ = E[f ], ν = E[g]. Observe that unlike the Theorem 3.1 [DMR06], we gain a better tradeo between and δ . We allow δ to be poly(), i.e. not too small, (instead of δ = exp(− 1 ) implicitly appearing in [DMR06]). On the other hand, we get a bound on hf, T ⊗n gi as a function of ρ0 ∈ (ρ, 1) instead of ρ. 43
9.1 Proof of the theorem Note that (9) follows from (10). Indeed, apply (10) to 1 − g to obtain
f, T ⊗n (1 − g) ≤ hFµ , Uρ0 F1−ν iγ + and then use the equalities
f, T ⊗n (1 − g) = hf, 1i − f, T ⊗n g = µ − f, T ⊗n g = hFµ , Uρ0 1iγ − f, T ⊗n g . So our goal in this section is to prove (10).
De nition 9.2 For a function f with range R de ne a function chop(f ) as f (x) f (x) ∈ [0, 1] chop(f )(x) = 0 f (x) < 0 1 f (x) > 1 The following theorem is proven in [MOO05]. It says that if a function f : [q]n → [0, 1] is smooth and all in uences are small, then fe and chop(fe) are close (where fe is as in De nition 8.8 with respect to some orthonormal basis {αi }). It essentially follows from what is known as the invariance principle, which says that the distribution of values obtained by f and that of values obtained by fe are close. Since f never deviates from [0, 1], it implies that fe rarely deviates from [0, 1].
Theorem 9.3 ([MOO05, Theorem 3.20]) Let q be a xed integer and {αi } be
some orthonormal basis of Rq . Then for every η < 1 and > 0 there is δ = O( 1−η ) that satis es the following: For a function f : [q]n → [0, 1] with decomposition as in (7) X 1
fˆ(αx )αx
f=
de ne fe : Rqn → R as in de nition 8.8 fe =
X
fˆ(αx )Γx
If f is η-smooth and all in uences are smaller than δ , i.e. ∀k
X
fˆ(αx )2 ≤ η k
and
∀i Inf i (f ) ≤ δ
x:|x|≥k
then
e
e
f − chop(f ) ≤
(where the constant in the O notation depends on q and {αi }, but not on n) 44
We also make use of the following powerful theorem by Borell. It says that the functions that maximize the inner product under the operator Uρ are the indicator functions of half-spaces.
Theorem 9.4 (Borell [Bor85]) Let
f, g : Rn → [0, 1] be two functions, and let
µ = Eγ [f ], ν = Eγ [g]. Then
f, Uρ⊗n g
γ
≤ hFµ , Uρ Fν iγ .
The above theorem only applies to the Ornstein-Uhlenbeck operator. A similar statement is derived for more general operators. The proof follows by writing a general operator as a product of the Ornstein-Uhlenbeck operator and some other operator.
Corollary 9.5 Let
f, g : R(q−1)n → [0, 1] be two functions and de ne µ = Eγ [f ]
and ν = Eγ [g]. For given 1 = λ0 > λ1 ≥ · · · ≥ λq−1 > −1 let ρ = max{|λ1 |, |λq−1 |} < 1. De ne T˜ = Uλ1 ⊗ Uλ2 ⊗ · · · ⊗ Uλq−1 . Then D
f, T˜⊗n g
E γ
Proof
≤ hFµ , Uρ Fν iγ .
For 1 ≤ i ≤ q − 1, let δi = λi /ρ. Note that |δi | ≤ 1 for all i. Let S be the
operator de ned by
S = Uδ1 ⊗ Uδ2 ⊗ · · · ⊗ Uδq−1 . Then,
Uρ⊗(q−1) S = Uρ Uδ1 ⊗ · · · ⊗ Uρ Uδq−1 = Uρδ1 ⊗ · · · ⊗ Uρδq−1 = T˜ ⊗(q−1)n ⊗n It follows that T˜⊗n = Uρ S . Since S ⊗n is an averaging operator, the func-
tion S ⊗n g obtains values in [0, 1] and satis es Eγ [S ⊗n g] = Eγ [g]. Thus D E
f, T˜⊗n g = f, Uρ⊗(q−1)n (S ⊗n g) γ
The corollary follows by applying Theorem 9.4 to the functions f and S ⊗n g . We are now ready to prove the rst step in the proof of Theorem 9.1. We use Theorem 9.3 and Corollary 9.5.
Lemma 9.6 Let
T be a symmetric linear operator on Rq with eigenvalues 1 =
λ0 > λ1 ≥ · · · ≥ λq−1 > −1 and corresponding orthonormal eigenvectors α0 = 1, α1 , . . . , αq−1 . Let T˜ = Uλ1 ⊗ Uλ2 ⊗ · · · ⊗ Uλq−1 . Let ρ = max{|λ1 |, |λq−1 |} < 1. 45
For any η < 1, > 0 there is δ = 1−η , where C that depends only on ρ, s.t. the following holds: For any two functions f, g : [q]n → [0, 1] with decomposition as in (7) if both are η -smooth, i.e. C
∀k
X
fˆ(αx )2 ≤ η k
and ∀k
x:|x|≥k
X
gˆ(αx )2 ≤ η k
x:|x|≥k
and all in uences in both of them are bounded by δ , i.e. ∀i Inf i (f ) < δ
and
∀i Inf i (g) < δ
then
f, T ⊗n g ≤ hFµ , Uρ Fν i +
where E[f ] = µ, E[g] = ν .
Proof
1
Apply Theorem 9.3 to get δ = δ(η, /4) = O( 1−η ) . Let µ0 = E[chop(fe)],
ν 0 = E[chop(e g )]. Then
|µ0 − µ| = |E[chop(fe) − fe]| = |hchop(fe) − fe, 1i| ≤ chop(fe) − fe ≤ /4 |ν 0 − ν| = |E[chop(e g ) − ge]| = |hchop(e g ) − ge, 1i| ≤ kchop(e g ) − gek ≤ /4 A straightforward calculation then show
|hFµ , Uρ Fν i − hFµ0 , Uρ Fν 0 i| ≤ /2 The required result follows by applying the corollary 9.5 D E
f, T ⊗n g = fe, T˜⊗n ge D E D E = chop(fe), T˜⊗n chop(e g ) + chop(fe), T˜⊗n (e g − chop(e g )) + D E + fe − chop(fe), T˜⊗n ge [Cauchy Schwartz] ≤ hchop(fe), T˜⊗n chop(e g )i + /4 + /4 [Cor. 9.5]
≤ hFµ0 , Uρ Fν 0 i + /2 ≤ hFµ , Uρ Fν i +
The following lemma proves Theorem 9.1. Let T be some xed Markov operator on [q] with eigenvalues 1 = λ0 > λ1 ≥ · · · ≥ λq−1 > −1 and corresponding orthonormal basis of eigenvectors α0 = 1, α1 , . . . , αq−1 . Assume that ρ = r(T ) < 1. 46
Recall that for any γ > 0 the linear operator Tγ on Rq is de ned by: Tγ 1 = 1 and Tγ v = γv for v ⊥ 1. It is easy to see that the operator S = T Tγ has the same eigenvectors as T and the corresponding eigenvalues are 1 = λ0 > λ1 γ ≥ · · · ≥
λq−1 γ > −1 (as long as γ < 1/ρ).
Lemma 9.7 Fix T as above and ρ0 ∈ (ρ, 1). For any > 0 there are δ = C and
k = C lg( 1 ), where C = C(ρ, ρ0 ) such that the following holds: If f, g : [q]n → [0, 1]
are two functions satisfying
≤k min Inf ≤k i (f ), Inf i (g) < δ
∀i
then
f, T ⊗n g ≤ hFµ , Uρ0 Fν iγ +
where µ = E[f ], ν = E[g].
Proof
Set η =
ρ ρ0
< 1 and denote S = T T 1 . Then S has the same eigenvectors η
as T , largest eigenvalue 1 and r(S) = ηρ = ρ0 < 1. We also denote X X |x| |x| f1 = T√η f = fˆ(αx )η 2 αx and g1 = T√η g = gˆ(αx )η 2 αx Using this notation it is easy to see that we can express hf, T ⊗n gi as
f, T ⊗n g = f1 , S ⊗n g1
(11)
We apply Lemma 9.6 with operator S and parameters η and /2 to get δ 0 = 1
δ9.6 (S, η, 2 )/2 = O( 1−η ) = poly(), where the degree of the polynomial depends only on ρ and ρ0 . Let k = O(lg( 1 )) be such that η k < min(δ 0 , /4), and let 0
δ = ( δ )2 = poly(). We show that these δ and k satisfy the requirements of the 8k lemma.
≤k We take two functions f and g such that ∀i min Inf ≤k i (f ), Inf i (g) < δ . Thus f1 and g1 are η -smooth and satisfy the same assumption. However, we cannot apply Lemma 9.6 on them with the operator S , as the requirement is that all in uences in both of them are small. In order overcome this problem we de ne two functions f2 and g2 with small in uences such that hf1 , S ⊗n g1 i ≈ hf2 , S ⊗n g2 i. De ne 0 Bf = {i : Inf ≤k i (f ) ≥ δ }
0 Bg = {i : Inf ≤k i (g) ≥ δ }
≤k Then |Bf |, |Bg | ≤ k/δ 0 . Moreover Bf ∩Bg = ∅ as δ < δ 0 and ∀i min(Inf ≤k i (f ), Inf i (g)) <
δ . We de ne f2 (y) and g2 (y) as the average over the coordinates in Bf and Bg 47
respectively, namely
X
f2 (y) = Eyi :i∈Bf [f1 (y)] =
|x| fˆ(αx )η 2 αx (y)
x:xBf =0
X
g2 (y) = Eyi :i∈Bg [g1 (y)] =
gˆ(αx )η
|x| 2
αx (y)
x:xBg =0
Clearly E[f2 ] = E[f ] = µ, E[g2 ] = E[g] = ν . We have Inf i (f2 ) = 0 for i ∈ Bf and k 0 Inf i (f2 ) ≤ Inf ≤k i (f ) + η < 2δ otherwise. Same holds for g2 . Their smoothness
follows from smoothness of f1 , g1 and we can apply Lemma 9.6 with the operator
S to get (12)
hf2 , S ⊗n g2 i ≤ hFµ , Uρ0 Fν i + /2 The only thing left to show is
| f1 , S ⊗n g1 − f2 , S ⊗n g2 | ≤ /2
(13)
Here we use the assumption that a coordinate cannot have a signi cant in uence of both functions.
! X Y
λ xi ⊗n ⊗n |x| ˆ | f1 , S g1 − f2 , S g2 | = η f (αx )ˆ g (αx ) x:xB ∪Bg 6=0 i:xi 6=0 η f X X |x| ˆ ≤ |fˆ(αx )ˆ g (αx )| + ρ | f (α )ˆ g (α ) x x x:|x|≤k x:xBf ∪Bg 6=0
[ρ < η ] ≤
X
x:|x|>k
X
|fˆ(αx )ˆ g (αx )| + η k
i∈Bf ∪Bg x:|x|≤k xi 6=0
[Cauchy Schwartz] ≤
X q
q k Inf ≤k (f ) Inf ≤k i i (g) + η
i∈Bf ∪Bg
√ [i ∈ Bf ⇒ Inf i (g) < δ] ≤ (|Bf | + |Bg |) δ + η k 2k δ 0 |Bf |, |Bg | ≤ k/δ 0 , η k ≤ /4 ≤ + /4 δ 0 8k = /2 Combining (11), (12) and (13) we get the required result
hf, T ⊗n gi ≤ hFµ , Uρ0 Fν i + and the proof of theorem is complete. We need the following corollary for our reduction in the next section. 48
Corollary 9.8 Let
q be a xed integer and T be a symmetric Markov operator
on [q] such that ρ = r(T ) < 1. Then for any > 0 there exist δ = poly() and k = O(lg( 1 )) such that the following holds: For any f, g : [q]n → [0, 1], if E[f ] > , E[g] > , and hf, T ⊗n gi = 0, then ∃i ∈ {1, . . . , n},
Proof
Denote ρ0 =
2ρ 1+ρ
Inf ≤k i (f ) ≥ δ
and
Inf ≤k i (g) ≥ δ .
(note that ρ < ρ0 < 1), and let 0 = hF , Uρ0 (1 − F1− )iγ .
Then
f, T ⊗n g < FE[f ] , Uρ0 (1 − F1−E[g] ) γ − 0
We apply the Theorem 9.1 to get δ = poly(0 ) and k = O(lg( 10 )), and some ≤k i ∈ {1, . . . , n} s.t. Inf ≤k i (f ) > δ and Inf i (g) > δ . The corollary follows from
equation (8), since 0 = hF , Uρ0 (1 − F1− )iγ = poly(), where the degree of the polynomial depends only on ρ.
10
Reduction
We are now ready to prove our main theorem. Our starting point is the following conjecture [Kho02]:
Conjecture 10.1 (Bipartite 2-to-1 Conjecture) For any
> 0 there is R =
R() s.t. the following problem is NP-hard. Given a bipartite 2-to-1 LC instance Φ = (V ∪W, E) with label sets {1, . . . , R} for V and {1, . . . , 2R} for W , distinguish
between the case where val(Φ) = 1 and the case where val(Φ) < .
In our case we make a stronger conjecture. We allow that label set to be of size O(lg(n)). The best possible soundness we could hope for is lg(n)−c for some constant c > 0. In addition, as a small technicality, we assume that all vertices of
W have the same degree. We restate the main theorem here for convenience.
Theorem 7.3
Assume that given a bipartite 2-to-1 Label Cover instance Φ with the label set of size R = O(lg(n)) it is NP-hard to distinguish between the case 1 where val(Φ) = 1 and the case where val(Φ) < f (n) , for some f (n). Then it is c NP-hard to color a 4-colorable graph with f (n) colors for some constant c > 0. The proof is by showing a reduction with the following properties: given an instance of 2-to-1 Label Cover Φ = (V ∪ W, E, R, Π) it produces a graph G on 49
|W | · 42R vertices. In the completeness part it is shown that if val(Φ) = 1 then G is 4-colorable. The soundness of the reduction says that if G contains an independent δ 2 k2 1 ≤ f (n)
set of size then val(Φ) ≥ Ω
= poly(), where δ and k are as in Corollary
9.8. In other words if val(Φ)
then χ(G) ≥ f c (n) for some constant c > 0.
10.1 Reduction De nition 10.2 We de ne a symmetric Markov operator T on {0, 1, 2, 3}2 such
that r(T ) < 1 and such that T ((x1 , x2 ) ↔ (y1 , y2 )) > 0 if and only if {x1 , x2 } ∩ {y1 , y2 } = ∅. Our operator has three types of transitions, with transitions probabilities β1 , β2 , and β3 . • With probability β1 we have (x, x) ↔ (y, y) where x 6= y . • With probability β2 we have (x, x) ↔ (y, z) where x, y, z are all dierent. • With probability β3 we have (x, y) ↔ (z, w) where x, y, z, w are all dierent.
For T to be a symmetric Markov operator, we need that β1 , β2 and β3 are nonnegative and 3β1 + 6β2 = 1,
2β2 + 2β3 = 1.
For example for β1 = 121 , β2 = 18 , and β3 = 38 we have ρ = r(T ) = 5/6
Reduction
We start with a 2-to-1 Label Cover instance Φ = ((V ∪W, E), R, 2R, Π).
Each (v, w) ∈ E is associated with a constraint πvw s.t. for each b ∈ 2R there is a unique a s.t. (a, b) ∈ πvw (we denote a = πvw (b)) and for each a ∈ R there are −1 exactly two b1 , b2 ∈ 2R s.t. (a, bi ) ∈ πvw (denote (b1 , b2 ) = πvw (a)). We construct
G0 = (V 0 , E 0 ) as follows: • Each vertex w ∈ W is replaced by a copy of {0, 1, 2, 3}2R (denote by [w]). S The set of vertices in G0 is V 0 = w∈W [w] = W × {0, 1, 2, 3}2R . • Let T be as in de nition 10.2. For every w1 , w2 ∈ W that have a common neighbor v ∈ V let π1 , π2 be the corresponding constraints. We set an edge between (w1 , x) and (w2 , y) if T (xπ1−1 (k) , yπ2−1 (k) ) 6= 0 for all k ∈ R, or equivalently {xi1 , xj1 } ∩ {xi2 , xj2 } = ∅ where π1−1 (k) = (i1 , j1 ) and π2−1 (k) = (i2 , j2 ).
50
Completeness
Assume there is a labeling L such that wL (Φ) = 1. Let c(w, x) =
xL(w) for all w ∈ W . We show that this is a legal coloring of G0 . Pick an edge ((w1 , x), (w2 , y)) ∈ E 0 . Then w1 , w2 have a common neighbor v ∈ V . Let π1 and π2 be the corresponding constraints, and let k = L(v). Then π1 (L(w1 )) = k = π2 (L(w2 )), as L satis es all the constraints. Since ((w1 , x), (w2 , y)) ∈ E 0 , the sets xπ1−1 (k) and yπ2−1 (k) are disjoint and hence
c(w1 , x) 6= c(w2 , y) as c(w1 , x) = xL(w1 ) ∈ xπ1−1 (k) and c(w2 , y) = yL(w2 ) ∈ yπ2−1 (k) .
Soundness S ⊆ V 0 s.t.
Assume that χ(G0 ) ≤ Q. Then G0 contains an independent set |S| |V 0 |
≥
1 Q
= 2·. Our goal is to show that is such case val(Φ) > poly().
Let J be a subset of W that have a non-negligible contribution to S
J = {w ∈ W :
[w] ∩ S > } [w]
Markov inequality implies |J| ≥ |W |. For each w ∈ J let fw : {0, 1, 2, 3}2R → {0, 1} be the indicator function of S , i.e. fw (x) = 1 i (w, x) ∈ S . Then E[fw ] > for such w's. Let δ and k be as in Corollary 9.8 applied on the operator T from de nition 10.2 with parameter . We can de ne a small set of labels for w.
L(w) = {i : Inf ≤2k > δ/2} i Observe that |L(w)| ≤
4k . δ
Now we want to give labels to neighbors of J in Φ. We
prove the following claim.
Claim 10.3 Let v ∈ N (J) and let w1 , w2 ∈ N (v) ∩ J . Let π1 , π2 be the correspond-
ing constraints. Then there are i ∈ Lw1 , j ∈ Lw2 s.t. π1 (i) = π2 (j).
Proof
Recall that fw 's are indicators of an independent (x) = 1 = set. Thus fw1 fw2 (y) implies that ((w1 , x), (w2 , y)) ∈ / E 0 . Therefore T xπ−1 (k) , yπ−1 (k) = 0 for 1
2
some k ∈ R and thus ⊗R T (xπ1−1 (1) , . . . , xπ1−1 (R) ), (yπ2−1 (1) , . . . , yπ2−1 (R) ) = 0 De ne
f (xπ1−1 (1) , . . . , xπ1−1 (R) ) = fw1 (x1 , . . . , x2R ) g(yπ2−1 (1) , . . . , yπ2−1 (R) ) = fw2 (y1 , . . . , y2R ) where we think of f , g as functions in R variables, each taking values in {0, 1, 2, 3}2 .
We show that f , T ⊗R g = 0. Then using Corollary 9.8 we conclude that there is 51
≤k ` ∈ R s.t. Inf ≤k l (f ) > δ and Inf l (g) > δ . Using the relation between f and fw1 ,
we conclude that there is some i ∈ π1−1 (`) such that Inf ≤2k (fw1 ) > δ/2. Similarly i for g there is some j ∈ π2−1 (`) such that Inf ≤2k (fw2 ) > δ/2. Therefore there are j
i ∈ Lw1 , j ∈ Lw2 s.t. π1 (i) = π2 (j).
So it is left to show that f , T ⊗R g = 0. And indeed: X X
1 f , T ⊗R g = 2R f (x) T ⊗R (x, y)g(y) 4 2 R 2 R x∈({0,1,2,3} )
y∈({0,1,2,3} )
X 1 X = 2R fw1 (x) T ⊗R (xπ1−1 , yπ2−1 )fw2 (y) 4 x y X 1 = 2R T ⊗R (xπ1−1 , yπ2−1 ) 4 x:fw1 (x)=1 y:fw2 (y)=1
=
1 42R
X
0
x:fw1 (x)=1 y:fw2 (y)=1
= 0 From the claim we get that for all v ∈ N (J) and any w1 , w2 ∈ N (v) ∩ J 2 1 δ Pr [π1 (i) = π2 (j)] ≥ ≥ i∈L(w1 ) |L(w1 )||L(w2 )| 4k j∈L(w2 )
By averaging there is L0 : V ∪ W → 2R such that
2 δ Pr [L0 (v) = π(L0 (w))|w ∈ J] ≥ v∈N (w) 4k Hence, if we assume regularity on the vertices of W , we get
Pr[L(v) = π(L(w))] ≥ Pr [w ∈ J] Pr [L0 (v) = π(L0 (w))|w ∈ J] ≥ vw
w∈W
v∈N (w)
δ 4k
2 = poly()
We conclude that the value of G is at least poly() which completes the soundness analysis of the reduction.
11
Conclusions and Final Remarks
We have shown a reduction that takes a 2-to-1 Label Cover instance Φ = (V ∪
W, E, R) and produces a graph G on |W | · 2O(R) vertices. If val(Φ) = 1 then G is 452
colorable, and if val(Φ) < , then α(G) < c and thus χ(G) >
1 c
for some c > 0.
Since our gadgets are of size 2O(R) and we wish to have a polynomial reduction, we bound R to be at most logarithmic in the size of Φ. The best we could hope for is = 1/ lgδ (n), and thus conclude the hardness of coloring a 4-colorable graph with lgc (n) colors. One possible extension of this result could be nding a family of smaller gadgets (and not exponential), so that the Majority is Stablest theorem can be applied on it. Then taking larger values of R will allow us to translate the soundness of 2-to-1 Label Cover problem into a stronger hardness of graph coloring, as the relative size of the independent set will be larger compared to the size of the entire graph.
53
54
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60
Appendix A
Minimum of Q(s)
The claims in the appendix deal with nding minimum of a certain given function. They correspond to Lemma 2.5 of [Fri08].
Claim A.1 Fix some natural t and r. De ne Q(s) on natural values of s to be Q(s) =
1 − r−1
s X t−1 s j t j · (−1) · (r − r ) j j=0
1. If r > t + 1, then mins≥1 {Q(s)} = −1, which is attained only at s = 1, 2 . . . t. 2. Moreover, since lims→∞ Q(s) = 0, the second minimal value is bounded above −1 by some η = η(r, t).
Proof
P j t j s Denote by P (s) the sum t−1 j=0 (−1) · (r − r ) j . We may think of P as a polynomial in reals, where xj = j!1 · x(x − 1) · · · (x − j + 1). That is P (x) is a s 1 polynomial of degree t − 1 and Q(s) = P (s) − r−1 for natural s. We rst write P (s) in a dierent form for s ∈ N: t−1 X
s P (s) = (−1) · (r − r ) j j=0 X t−1 t−1 X t j s j s = r (−1) − (−r) j j j=0 j=0 Recalling that also write
Ps
j j=0 (−1)
s j
j
t
j
= (1 − 1)s = 0 and s X
Ps
j j=0 (−r)
s j
= (1 − r)s , we can
s P (s) = −(1 − r) + (−1) (r − r ) j j=t+1 s
j
j
t
In this form it is easy to see that indeed P (s) = −(1 − r)s for s = 1, 2 . . . t, which proves the equality part of item 1.
t t+1 r (r − 1) − (r − 1)
Some more tedious computation give P (t + 1) = (−1) t t t+2 and P (t + 2) = (−1) r (r − 1)(r − t − 1) − (r − 1) . Therefore Q(t + 1) = t t r−t−1 r r − 1 > 0 and Q(t + 2) = r−1 − 1. For r ≥ t + 2 we get Q(t + 2) > −1. r−1 r−1 t+1
61
In order to show that Q(s) > −1 for s > t + 2 it is enough to show that
|P (x)| < (r − 1)x for all x > t + 2. Since P (i) = (1 − r)i for i = 1, . . . t, it changes its sign t − 1 times in the interval
(1, t) and thus can be written as t−1
P (x) = (−1)
t−1 Y (x − ai )
ai ∈ (i, i + 1)
where
i=1
Therefore P (x) doesn't change its sign for x > t. Let's assume that P (t) > 0, i.e.
t is odd (same proof works for even t). We derive P to get for x > t P 0 (x) = P (x) ·
t−1 X i=1
1 x − ai
In particular for x ≥ t + 2 0
P (x) = P (x) ·
t−1 X i=1
1 ≤ P (x) x − ai
1 1 1 + + ... 2 3 t
< P (x) ln(t) ≤ P (x) ln(r − 1)
This implies that for x ≥ t + 2
ln(P (x))0 =
P 0 (x) < ln(r − 1) = ln((r − 1)x )0 P (x)
and together with ln(P (t + 2)) < ln((r − 1)t+2 ) we conclude that for x ≥ t + 2
ln(P (x)) < ln((r − 1)x )
Therefore P (x) < (r − 1)x for all x ≥ t + 2, and the proof of the claim is complete.
The following claim is a generalization of the previous one for the case where the eigenvalues are not equal.
Claim A.2 Fix some natural t and r. Let 1 = λ0 > λ1 ≥ · · · ≥ λr−1 =: λ∞ > −1
and λS = ni=1 λSi for S ∈ [r]n . Assume λ∞ < 0, and let γ = 1 − λ1∞ > 1. De ne Q(S) : [r]n → R to be Q
t−1 X
|S| Q(S) = λS (−1) (γ − γ ) i j=0
1.
λ1 −λ∞
j
t
j
n o 1 < min 1, (1−λ∞ then minS6=0 {Q(S)} = −1, which is attained only t ) −1
at S that satisfy |S| = 1 . . . t and λS = (λ∞ )|S| . 62
2. Moreover, since lim|S|→∞ Q(S) = 0, the second minimal value is bounded away from −1 by some function of γ and t.
Proof
The proof is similar to the previous claim by writing Q(S) = λS P (|S|),
where P is the polynomial part of Q. That is t−1 X
s P (s) = (−1) · (γ − γ ) j j=0 j
t
j
which also equals s X
s P (s) = −(1 − γ) + (−1) (γ − γ ) j j=t+1 s
j
Then P (s) = −(1 − γ)s for s = 1 . . . t and |S| Y λS 1 i Q(S) = −λS = λ∞ λ∞
j
t
for 1 ≤ |S| ≤ t
Si 6=0
Since λ1 < −λ∞ we conclude that Q(S) ≥ −1 for |S| = 1 . . . t. with equality i
λS = (λ∞ )|S| . |S| = t + 1. As in the previous claim P (t + 1) = (−1)t+1(γ − Next we consider Q 1) γ t − (γ − 1)t . If λ1 ≤ 0, then Q(S) = λS · P (S) = (−λ ) (γ − 1) γ t − Si Si 6=0 t (γ − 1) ≥ 0. Otherwise, for λ1 > 0, it is easy to see that the function Q(S) is minimized by S such that λS = λ1 · (λ∞ )t and
t t min {Q(S)} = −λ1 (−λ∞ ) (γ − 1) γ − (γ − 1) t
S:|S|=t+1
=
λ1 (1 − λ∞ )t − 1 λ∞
which is larger that -1 by the assumption
λ1 −λ∞ s
<
1 (1−λ∞ )t −1
For s ≥ t + 2 we have |P (s)| < (γ − 1) . Thus for |S| ≥ t + 1 holds |Q(S)| < |S|
|λS |(γ − 1)|S| ≤ λ∞ (γ − 1)|S| = 1. Therefore Q(S) > −1 for |S| ≥ t + 2.
63
