The Average-Case Complexity of Counting Cliques in Erdős-Rényi Hypergraphs

Abstract

The complexity of clique problems on Erds-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 Erds-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 prove the following: •Dense Erds-Rényi 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 nΩ(k) time to count k-cliques in G(n, c, s) if k and c are constant. • Sparse Erds-Rényi hypergraphs: When c = Θ(n-α), for each fixed α 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 ωk/3-C α (k/2) + O-k, α (1), where ω/9 ≤ C ≤ 1 and ω is the matrix multiplication constant. In the hypergraph case of s ≥ 3, we show a lower bound at the exponent of k-α (k/s) + O-k, α (1) which surprisingly is tight against algorithmic achievability exactly for the set of c above the Erds-Rényi k-clique percolation threshold. Our reduction yields the first known average-case hardness result on Erdos-Renyi hypergraphs based on a worst-case hardness assumption. 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 = Θ(n-α).