The Monotone Complexity of k-Clique on Random Graphs

It is widely suspected that Erdos-Renyi 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 G(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 quasi-sunflowers, a new relaxation of sunflowers in which petals may overlap slightly on average. A "quasi-sunflower lemma" (a la the Erdos-Rado sunflower lemma) leads to our novel lower bounds within Razborov's method of approximations.