Clique Is Hard on Average for Regular Resolution

Abstract

We prove that for $k\ll \sqrt[4]{n}$ regular resolution requires length $n^{\Omega (k)}$ to establish that an Erdős-Rényi graph with appropriately chosen edge density does not contain a $k$-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional $n^{\Omega (k)}$ lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.