The average-case complexity of counting cliques in Erdős-Rényi Hypergraphs

Abstract

The complexity of clique problems on Erdős-Rényi random graphs has become a central topic in average-case complexity. Algorithmic phase transitions in these problems have been shown to have broad connections ranging from mixing of Markov chains and statistical physics to information-computation gaps in high-dimensional statistics. We consider the problem of counting k-cliques in s-uniform Erdős-Rényi hypergraphs G(n, c, s) with edge density c and show that its fine-grained average-case complexity can be based on its worst-case complexity. We give a worst-case to average-case reduction for counting k-cliques on worst-case hypergraphs given a blackbox solving the problem on G(n, c, s) with low error probability. Our approach is closely related to [Goldreich and Rothblum, FOCS18], which showed a worst-case to average-case reduction for counting cliques for an efficiently-sampleable distribution on graphs. Our reduction has the following implications: --

Dense Erdős-Rényi graphs and hypergraphs: Counting k-cliques on G(n, c, s) with k and c constant matches its worst-case complexity up to a polylog(n) factor. Assuming ETH, it takes n(superscript [omega](k) time to count k-cliques in G(n, c, s) if k and c are constant. --

Sparse Erdős-Rényi graphs and hypergraphs: When c = [theta](n⁻[superscript alpha]), for each fixed [alpha] our reduction yields different average-case phase diagrams depicting a tradeoff between runtime and k. Assuming the best known worst-case algorithms are optimal, in the graph case of s = 2, we establish that the exponent in n of the optimal running time for k-clique counting in G(n, c, s) is wk/3 - C[alpha](k 2) + [omicron][subscript k,[alpha]](1), where w/9 </= C </= 1 and w is the matrix multiplication constant. In the hypergraph case of s >/= 3, we show a lower bound at the exponent of k - [alpha](k s) + [omicron][subscript k,[alpha]](1) which surprisingly is tight against algorithmic achievability exactly for the set of c above the Erdős-Rényi k-clique percolation threshold. Our reduction yields the first average-case hardness result for a problem over Erdős-Rényi hypergraphs based on a corresponding worst-case hardness assumption.

Moreover, because we consider sparse Erdős-Rényi hypergraphs, for each n, k, and s we actually have an entire family of problems parametrized by the edge probability c and the behavior changes as a function of c; this is the first worst-to-average-case hardness result we are aware of for which the complexity of the same problem over worst-case versus average-case inputs is completely different. We also analyze several natural algorithms for counting k-cliques in G(n, c, s) that establish our upper bounds in the sparse case c = [theta](n⁻[superscript alpha]). The ingredients in our worst-case to average-case reduction include: (1) several new techniques for the self-reducibility of counting k-cliques as a low-degree polynomial; and (2) a finite Fourier analytic method to bound the total variation convergence of random biased binary expansions to the uniform distribution over residues in F[subscript p].