The Monotone Complexity of k-clique on Random Graphs

It is widely suspected that Erd\H{o}s-R\'enyi random graphs are a source of hard instances for clique problems. Giving further evidence for this belief, we prove the first average-case hardness result for the $k$-clique problem on monotone circuits. Specifically, we show that no monotone circuit of size $O(n^{k/4})$ solves the $k$-clique problem with high probability on $\ER(n,p)$ for two sufficiently far-apart threshold functions $p(n)$ (for instance $n^{-2/(k-1)}$ and $2n^{-2/(k-1)}$). Moreover, the exponent $k/4$ in this result is tight up to an additive constant. One technical contribution of this paper is the introduction of {\em quasi-sunflowers}, a new relaxation of sunflowers in which petals may overlap slightly on average. A ``quasi-sunflower lemma'' (\`a la the Erd\H{o}s-Rado sunflower lemma) leads to our novel lower bounds within Razborov's method of approximations.

[1]  Ari Juels,et al.  Hiding Cliques for Cryptographic Security , 1998, SODA '98.

[2]  Paul Beame.  Lower bounds for recognizing small cliques on CRCW PRAM's , 1990, Discret. Appl. Math..

[3]  Benjamin Rossman,et al.  On the constant-depth complexity of k-clique , 2008, STOC.

[4]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[5]  Ran Raz,et al.  Higher lower bounds on monotone size , 2000, STOC '00.

[6]  Akira Maruoka,et al.  A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with at most (1/6)log log n Negation Gates , 2005, SIAM J. Comput..

[7]  Svatopluk Poljak,et al.  On the complexity of the subgraph problem , 1985 .

[8]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[9]  Kazuyuki Amano,et al.  k-Subgraph Isomorphism on AC0 Circuits , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[10]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[11]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[12]  Alexander A. Razborov,et al.  On the method of approximations , 1989, STOC '89.

[13]  J. Spencer.  Intersection Theorems for Systems of Sets , 1977, Canadian Mathematical Bulletin.

[14]  Stasys Jukna,et al.  Extremal Combinatorics - With Applications in Computer Science , 2001, Texts in Theoretical Computer Science. An EATCS Series.

[15]  Svante Janson,et al.  Poisson Approximation for Large Deviations , 1990, Random Struct. Algorithms.

[16]  Noga Alon,et al.  Finding a large hidden clique in a random graph , 1998, SODA '98.

[17]  Benjamin Rossman,et al.  Average-case complexity of detecting cliques , 2010 .

[18]  Benjamin Rossman.  The Monotone Complexity of k-clique on Random Graphs , 2010, FOCS.

[19]  Béla Bollobás,et al.  Random Graphs: Preface to the Second Edition , 2001 .

[20]  Noga Alon,et al.  The Probabilistic Method, Third Edition , 2008, Wiley-Interscience series in discrete mathematics and optimization.

[21]  Akira Maruoka,et al.  The Potential of the Approximation Method , 2004, SIAM J. Comput..

[22]  Coenraad Bron,et al.  Finding All Cliques of an Undirected Graph (Algorithm 457) , 1973, Commun. ACM.

[23]  B. Bollobás.  Threshold functions for small subgraphs , 1981 .

[24]  Akira Maruoka,et al.  A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with At Most (1/6) log log n Negation Gates , 1998, MFCS.