It is known that, for every constant $kgeq 3$, the presence of a
  $k$-clique (a complete subgraph on $k$ vertices) in an $n$-vertex
  graph cannot be detected by a monotone boolean circuit using fewer
  than $Omega((n/log n)^k)$ gates.  We show that, for every constant
  $k$, the presence of an $(n-k)$-clique in an $n$-vertex graph can be
  detected by a monotone circuit using only $O(n^2log n)$ gates.
  Moreover, if we allow unbounded fanin, then $O(log n)$ gates are
  enough

Andreev, Alexander E.

Jukna, Stasys

It is known that, for every constant k ≥ 3, the presence of a k-clique (a complete subgraph on k vertices) in an n-vertex graph cannot be detected by a monotone boolean circuit using fewer than Ω((n / log n)k) gates. We show that, for every constant k, the presence of an (n − k)-clique in an n-vertex graph can be detected by a monotone circuit using only O(n2 log n) gates. Moreover, if we allow unbounded fanin, then O(log n) gates are enough

A. E. Andreev

S. Jukna

CiteSeerX

Very Large Cliques are Easy to Detect

Dagstuhl Research Online Publication Server

Very Large Cliques are Easy to Detect ∗(Preliminary Version)A.E. AndreevLSI Logic Corporation, AE-1871551 McCarthy Blvd.CA 95305 Milpitas, USAandreev@lsil.comS. JuknaInstitute of MathematicsAkademijos 4LT-08663 Vilnius, LithuaniaMay 25, 2006AbstractIt is known that, for every constant k ≥ 3, the presence of a k-clique (acomplete subgraph on k vertices) in an n-vertex graph cannot be detectedby a monotone boolean circuit using fewer than Ω((n/ log n)k) gates. Weshow that, for every constant k, the presence of an (n− k)-clique in an n-vertex graph can be detected by a monotone circuit using only O(n2 log n)gates. Moreover, if we allow unbounded fanin, then O(log n) gates areenough.Keywords: Clique function, monotone circuits, perfect hashing, ver-tex cover, critical graphsAMS subject classification: 05C35, 05C60, 68Q17, 68R10, 94C301 IntroductionWe consider the well-known Clique function CLIQUE(n, s). This is a monotoneboolean function on(n2)boolean variables representing the edges of an undi-rected graph G on n vertices, whose value is 1 iff G contains an s-clique. Weare interested in proving good upper bounds on the size of monotone circuitswith fanin-2 AND and OR gates computing CLIQUE(n, s).The only non-trivial upper bound for CLIQUE(n, s) we are aware of is a non-monotone upper bound O(n2.5ds/3e) given in [3, 10] (see also [1]). This boundis obtained by a reduction to boolean matrix multiplication. Until recently, nomonotone circuits better than DNFs for this function were known.A trivial depth-2 formula—its minimal DNF—has(ns)−1 fanin-2 OR gates.Can we reduce the number of gates by allowing larger depth? In particular,can this number be made polynomial in n for growing s?∗The research reported herein was initiated by the discussions during the Dagstuhl-Seminar“Complexity of Boolean Functions” (March 2006).Dagstuhl Seminar Proceedings 06111Complexity of Boolean Functionshttp://drops.dagstuhl.de/opus/volltexte/2006/6092That it is impossible to save even one OR gate using so-called multilinearmonotone circuits—where inputs to each AND gate are computed from disjointsets of variables—was recently shown by Krieger in [8]: for any s, multilinearmonotone circuits for CLIQUE(n, s) require(ns)− 1 OR gates—just as many asthe minimal DNF of this function! That substantial savings are impossible evenin the class of all monotone circuits follows from well known lower bounds on themonotone circuit complexity of the clique function obtained by Razborov [12]and numerically improved by Alon and Boppana [1]: for every constant s ≥ 3,the function CLIQUE(n, s) cannot be computed by a monotone circuit usingfewer than Ω((n/ log n)s) gates, and for growing s we need at least 2Ω(√s) gates,as long as s ≤ (n/ log n)2/3/4.By a padding argument, this implies that even CLIQUE(n, n− k) requiressuper-polynomial number of gates, as long as k ≤ n/2 grows faster than log3 n.To see this, let m be the maximal number such that m − s ≤ k where s =(m/ logm)2/3/4. Then s = Ω(k2/3) and CLIQUE(m, s) is a sub-function of(i.e. can be obtained by setting to 1 some variables in) CLIQUE(n, n−k): justconsider only the n-vertex graphs containing a fixed clique on n −m verticesconnected to all the remaining vertices (the rest may be arbitrary). Thus,the function CLIQUE(m, s), and hence also the function CLIQUE(n, n − k),requires at least 2Ω(√s) = 2Ω(k1/3) gates, which is super-polynomial (in n) fork = ω(log3 n).But what is the complexity of CLIQUE(n, n − k) when k is indeed small,say, constant—can then this function be computed by a monotone circuit usingmuch fewer than(nk)OR gates?For k = 1 this was recently answered affirmatively by Krieger in [8]: thefunction CLIQUE(n, n−1) can be computed by a monotone ΠΣΠ-formula usingonly O(log n) OR gates. (Note that a DNF for this function has n−1 OR gates.)The argument of [8] uses the existence of particular error-correcting codes toencode (n− 1)-cliques, and does not seem to work for k > 1.In this paper we use another argument (based on perfect hashing) to obtaina more general result: a logarithmic number of OR gates is enough for everyconstant k, and a polynomial number of gates is enough also for growing k,as long as k = O(√log n). Moreover, we can define the desired ΠΣΠ-formulasexplicitely.2 ResultsTheorem 2.1. For every constant k, the function CLIQUE(n, n − k) can becomputed by a monotone ΠΣΠ-formula containing at most O(log n) OR gates.This theorem is a direct consequence of the following more general resultwhich, for every constant k, allows us also to explicitely construct such a for-mula.Recall that a vertex cover in a graph H is a set of its vertices containingat least one endpoint of each edge. The vertex cover number of H, denotedby τ(H), is the minimum size of such a set. A graph is τ -critical if removalof any its edges reduces the vertex cover number. For example, there are only3two τ -critical non-isomorphic graphs H with τ(H) = 2, a triangle and a graphconsisting of two disjoint edges. Erdo˝s, Hajnal and Moon [4] prove that everyτ -critical graph has at most(τ(H)+12)edges.In what follows, let G(r, k) denote the set of all τ -critical graphs on [r] ={1, . . . , r} with τ(H) = k + 1.Given a family F of functions f : [n]→ [r], where [n] = {1, . . . , n}, considerthe following monotone ΠΣΠ-formulaΦF (X) =∧H∈G(r,k)∧f∈F∨{a,b}∈E(H)∧e∈f−1(a)×f−1(b)xe.This formula rejects a given graph G = ([n], E) iff there exists a graph H ∈G(r, k) and a function f ∈ F such that for each edge {a, b} of H there is at leastone edge in the complement G of G between f−1(a) and f−1(b). The formulaΦF (X) has at most(|G(r, k)|+ |F |)(k + 22)≤ 2O(k2 log(r/k)) +O(k2|F |)OR gates, which is linear in |F | if both r and k are constants.A family F of functions f : [n] → [r] is s-perfect (n, r ≥ s) if for everysubset S ⊆ [n] of size |S| = s there is an f ∈ F such that |f(S)| = |S|. Thatis, for every s-element subset of [n] at least one function in F is one-to-onewhen restricted to this subset. Such families are also known in the literature as(n, r, s)-perfect hash families.Theorem 2.2. Let F be a (n, r, s)-perfect hash family with s = 2(k+1). Thenthe formula ΦF (X) computes CLIQUE(n, n− k).Using a simple probabilistic argument, Mehlhorn [9] shows that (n, r, s)-perfect hash families F of size |F | ≤ ses2/r log n exist. Thus, taking r = s, weobtain that for every k, a desired monotone ΠΣΠ-formula for CLIQUE(n, n−k)withϕ(n, k) = 2O(k2) +O(k2e2k log n)OR gates exists. This is O(log n) for every constant k, and polynomial fork = O(√log n). Recall that already for k = ω(log3 n), any monotone circuit forCLIQUE(n, n− k) requires a super-polynomial number of gates.Remark 2.3. In the class of ΠΣΠ-formulas, the upper bound ϕ(n, k) can-not be substantially improved because, as shown by Radhakrishnan [11], every(not necessarily monotone) ΠΣΠ-formula for the threshold function Tnn−k withk < (log log n)2 requires at least 2Ω(√k/ ln k) log n OR gates. This lower boundimplies the same lower bound for CLIQUE(n, n−k), because Tns is a monotoneprojection of CLIQUE(n, s): just assign all variables xe of CLIQUE(n, s) withi ∈ e the value of the i-th variable of Tns , that is, identify complete stars withtheir centers.4Using explicit (n, r, s)-perfect hash families we can obtain explicit formulas.For any constant s, (n, r, s)-perfect hash families F with |F | = O(logs n) canbe obtained by the following simple construction.Let M = {ma,i} be an n × b matrix with b = dlog ne whose rows aredistinct 0-1 vectors of length b. Let F = {f1, . . . , fb} be the family of functionsfi : [n] → {0, 1} determined by the columns of M ; hence, fi(a) = ma,i. Let Gbe an arbitrary (s + 1)-perfect family of functions g : {0, 1}s → [r]. Bondy’stheorem [2] says that the projections of any set of s+1 distict binary vectors onsome set of s coordinates must all be distinct. Hence, for any set a1, . . . , as+1of s + 1 rows there exist s columns fi1 , . . . , fis such that all s + 1 vectors~vj = (fi1(aj), . . . , fis(aj)), j = 1, . . . , s + 1 are distinct. Since the family G is(s+1)-perfect, at least one function g ∈ G will take different values on all theses + 1 vectors. Hence, the function h(x) = g(fi1(x), . . . , fis(x)) takes differentvalues on all s+1 points a1, . . . , as+1, as desired. Thus, taking the superpositionof functions from G with s-tuples of functions from F , we obtain a family H of|H| ≤(|F |s)· |G| = O(|G| logs n)functions h : [n]→ [r] which is (s+ 1)-perfect.If s is constant then, for example, we can take r = 2s and let G consistof the single function g(x) =∑si=1 xi2i−1. Then |H| = O(logs n). To makethe range size r smaller one can use, for example, the fact, due to Fredman,Komlo´s and Szemere´di [5], that if p is a prime larger than m, then the func-tions g1, g2, . . . , gp−1 with gα(x) = (αx mod p) mod r form a family of per-fect (m, r, s)-hash functions for every r ≥ s2. Using this fact, we can reducethe range size r till r = s2 at the cost of increasing the size of the family Gtill |G| = O(2k).Anyway, for constant k, Theorem 2.2 and our construction yieldsCorollary 2.4. For every constant k, there is an explicit monotone ΠΣΠ-formula for CLIQUE(n, n− k) using only O(log2k+2 n) OR gates.Remark 2.5. For fixed values of r and s, infinite classes of (n, r, s)-perfect hashfamilies F even with |F | = O(log n) were constructed by Wang and Xing in [13]using algebraic curves over finite fields. Using this (more involved) constructionone can achieve the upper bounds stated in Theorem 2.1 by explicit monotoneΠΣΠ-formulas.Remark 2.6. Let k be an arbitrary constant. In the proof of Theorem 2.2we construct a monotone ΣΠΣ-formula with O(log n) OR gates for the dualfunction of CLIQUE(n, n − k). (Recall that a dual of a boolean functionf(x1, . . . , xn) is the function f∗(x1, . . . , xn) = ¬f(¬x1, . . . ,¬xn), where “¬”denotes negation.) Moreover, this formula is multilinear, i.e. inputs to eachits AND gate are computed from disjoint sets of variables. On the other hand,Krieger [8] shows that every monotone multilinear circuit for CLIQUE(n, n−k)requires at least(nk)−1 OR gates. This gives an example of a boolean function,whose dual requires much larger multilinear circuits than the function itself.53 Proof of Theorem 2.2Instead of the function CLIQUE(n, n− k) it will be convenient to consider thedual function CLIQUE∗(n, n−k). Note that this function accepts a given graphG = ([n], E) iff G has no independent set with n−k vertices, which is equivalentto τ(G) ≥ k + 1. Hence, the graphs in G(n, k) are the smallest (with respectto the number of edges) graphs accepted by CLIQUE∗(n, n − k). Recall thatG(n, k) consists of all τ -critical graphs on [r] = {1, . . . , r} with τ(H) = k + 1.We will construct a monotone ΣΠΣ-formula for CLIQUE∗(n, n−k). ReplacingOR gates by AND gates (and vice versa) in this formula we obtain a monotoneΠΣΠ-formula for CLIQUE(n, n− k).Important for our construction is that the number of non-isolated verticesin graphs H ∈ G(n, k) depends only on k, and not on n. This is a directconsequence of a result, due to Hajnal [6], that in a τ -critical graph withoutisolated vertices every independent set of size s has at least s neighbors. (Forcompleteness, we include a short proof of this interesting result in the appendix.)Claim 3.1. Every graph in G(n, k) has at most s = 2(k + 1) non-isolatedvertices.Proof. Let G = (V,E) be a τ -critical graph without isolated vertices whichcannot be covered by k vertices. Since G is minimal, it can be covered by someset S of |S| = k + 1 vertices and by no smaller set. Hence the complementT = V − S is an independent set. By Hajnal’s theorem, the set T must haveat least |T | neighbors. Since all these neighbors must lie in S, the desiredupper bound |V | = |S|+ |T | ≤ 2|S| ≤ 2(k + 1) on the total number of verticesfollows.Let now F be an arbitrary s-perfect family of functions f : [n] → [r], andconsider the following monotone ΣΠΣ-formulaΦ∗(X) =∨H∈G(r,k)∨f∈FKf,H(X),whereKf,H(X) =∧{a,b}∈E(H)∨e∈f−1(a)×f−1(b)xe.To verify that this formula computes CLIQUE∗(n, n − k), is enough to showthat:(i) τ(G) ≥ k + 1 for every graph G accepted by Φ∗(X), and(ii) Φ∗(X) accepts all graphs from G(n, k).To show (i), suppose that Φ∗(X) accepts some graph G. Then this graphmust be accepted by some sub-formula Kf,H with f ∈ F and H ∈ G(r, k).That is, for every edge {a, b} in H there must be an edge in G joining somevertex i ∈ f−1(a) with some vertex j ∈ f−1(b). Hence, if a set S coversthe edge {i, j}, then the set f(S) must cover the edge {a, b}. Thus, if S is a6minimal vertex cover in G, then f(S) is a vertex cover in H, implying thatτ(G) = |S| ≥ |f(S)| ≥ τ(H) = k + 1.To show (ii), take an arbitrary graph G = ([n], E) in G(n, k). By Claim 3.1,G has at most s non-isolated vertices. By the definition of F , some function f :[n]→ [r] must be one-to-one on these vertices. Consider the graph H = ([r], E′)with {a, b} ∈ E′ iff {f−1(a), f−1(b)}∩E 6= ∅. Since G ∈ G(n, k) and f is one-to-one on all non-isolated vertices of G, the graph H belongs to G(r, k). Moreover,for every edge {a, b} of H, the pair e = {i, j} with f(i) = a and f(j) = b isan edge of G, implying that xe = 1. This means that the sub-formula Kf,H ofΦ∗(X), and hence, the formula itself must accept G.This completes the proof of Theorem 2.2.AcknowledgmentWe are thankful to Matthias Krieger for interesting discussions.References[1] N. Alon, R. Boppana, The monotone circuit complexity of Boolean func-tions, Combinatorica, 7(1) (1987), 1–22.[2] J. A. Bondy, Induced subsets, J. Combin. Theory (B), 12 (1972), 201–202.[3] F. Chung, R.M. Karp, in: Open problems proposed at the NSF Conf. onComplexity Theory, Eugene, Oregon 1984, SIGACT News 16 (1984), 46.[4] P. Erdo˝s, A. Hajnal, J. Moon, Mathematical notes, Am. Math. Monthly,71 (1964), 1107–1110.[5] M. Fredman, J. Komlo´s, E. Szemere´di, Storing a sparse table with O(1)worst-case access time, J. of the ACM, 31(3) (1984), 538–544.[6] A. Hajnal, A theorem on k-saturated graphs, Canad. Math. J. 17 (1965)720–724.[7] L. Lova´sz, Combinatorial Problems and Exercises, North-Holland, 1979.[8] M. Krieger, On the incompresability of monotone DNFs, in: Proc. 15th Int.Symp. on Fundamentals of Computation Theory (FCT’05), Liskiewicz, M.,Reischuk, R. (Eds.), Lect. Notes in Comput. Sci., vol. 3623, Springer-Verlag(2005), pp. 32–43. Journal version to appear in: Theory of ComputingSystems.[9] K. Mehlhorn, On the program size of perfect and universal hash functions,in: Proc. 23rd Annual IEEE Symposium on Foundations of ComputerScience (1982) 170–175.[10] J. Nesˇetrˇil, S. Poljak, On the complexity of the subgraph problem, CMUC26 (1985) 415–419.7[11] J. Radhakrishnan, ΣΠΣ threshold formulas, Combinatorica 14(3) (1994),345–374.[12] A.A. Razborov, Lower bounds on the monotone complexity of someBoolean functions, Soviet Math. Doklady 31 (1985) 354–357.[13] H. Wang, C. Xang, Explicit constructions of perfect hash families fromalgebraic curves over finite fields, J. Comb. Theory (A) 93 (2001) 112–124.4 AppendixTheorem 4.1 (Hajnal [6]). In a τ -critical graph without isolated vertices, everyindependent set of size s has at least s neighbors.Proof. (due to Lova´sz [7]) Let G = (V,E) be a τ -critical graph without isolatedvertices. Then G is also α-critical in that removal of any its edge increases itsindependence number α(G), i.e. the maximum size of an independent set in G.An independent set T is maximal if |T | = α(G).Let us first show that every vertex belongs to at least one maximal inde-pendent set but not to all such sets. For this, take a vertex x and an edgee = {x, y}. Remove e from G. Since G is α-critical, the resulting graph hasan independent set T of size α(G) + 1. Since T was not independent in G,x, y ∈ T . Then T −{x} is an independent set in G of size |T −{x}| = α(G), i.e.is a maximal independent set avoiding the vertex x, and T − {y} is a maximalindependent set containing x.Hence, if X is an arbitrary independent set in G, then the intersection of Xwith all maximal independent sets in G is empty. It remains therefore to showthat, if Y is an arbitrary independent set, and S is an intersection of Y withan arbitrary number of maximal independent sets, then|N(Y )| − |N(S)| ≥ |Y | − |S|,where N(Y ) is the set of all neighbors of Y , i.e. the set of all vertices adjacentto at least one vertex in Y . Since an intersection of independent sets is anindependent set, it is enough to prove the claim for the case when T is amaximal independent set and S = Y ∩ T . Since clearly N(S) ⊆ N(Y )− T , wehave|N(Y )| − |N(S)| ≥ |N(Y ) ∩ T | = |T | − |S| − |T − Y −N(Y )|= α(G)− |S|+ |Y | − |(T ∪ Y )−N(Y )| ≥ |Y | − |S|,where the last inequality holds because the set (T ∪Y )−N(Y ) is independent.

http://drops.dagstuhl.de/opus/volltexte/2006/609/pdf/06111.JuknaStasys.Paper.609.pdf

Very Large Cliques are Easy to Detect

Abstract

Similar works

Full text

Available Versions

CiteSeerX

Dagstuhl Research Online Publication Server