Abstract

Understanding the average-case complexity of natural problems on natural distributions is an important challenge for computational complexity. In this paper, we consider the average-case monotone complexity of the k-clique problem (for constant k) on Erdős-Rényi random graphs G(n, p). Our main result is a lower bound of ω(n k/4) on the size of monotone circuits which solve the k-clique problem asymptotically almost surely on G(n, p) for all functions p: N − → [0, 1] (or even just for two sufficiently far-apart threshold functions, such as n −2/(k−1) and 2n −2/(k−1)). While stronger lower bounds of ˜ Ω(n k) are known [17] for the worst-case monotone complexity of k-clique, our result is the first average-case monotone lower bound. This lower bound also supports the intuition that random graphs at the threshold are a source of hard instances for the k-clique problem. A further result of this paper is a nearly matching upper bound of n k/4+O(1) , obtained by monotonizing a construction of constant-depth circuits due to Amano [3]. This upper bound points out a gap between the worst-case and average-case monotone complexity of the k-clique problem. Similar bounds on the average-case complexity of k-clique for non-monotone constant-depth (AC 0) circuits were previously obtained by the author [18] (ω(n k/4) lower bound) and Amano [3] (n k/4+O(1) upper bound). We remark that the monotone lower bound of the present paper uses entirely different techniques from the AC 0 lower bound of [18]. In particular, we introduce a new variant of sunflowers and prove an analogue of the sunflower lemma that may be of independent interest. 1 1

Citations