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

The complexity of clique problems on Erdos-Renyi 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 to information-computation gaps in high-dimensional statistics. We consider the problem of counting k-cliques in s-uniform Erdos-Renyi 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: 1. Dense Erdos-Renyi 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^Ω(k) time to count k-cliques in G(n, c, s) if k and c are constant. 2. Sparse Erdos-Renyi hypergraphs: When c = Θ(n^-α), our reduction yields different average-case phase diagrams depicting a tradeoff between runtime and k for each fixed α. 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 ω k/3 - C αk2 + 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 - αks + O_k, α(1) which surprisingly is tight exactly for the set of c above the Erdos-Renyi k-clique percolation threshold. Our reduction yields the first known average-case hardness result on Erdos-Renyi hypergraphs based on worst-case hardness conjectures. 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).

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