The Average-Case Complexity of Counting Cliques in Erd\H{o}s-R\'enyi 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 $\mathrm{polylog}(n)$ factor. Assuming ETH, it takes $n^{\Omega(k)}$ time to count $k$-cliques in $G(n, c, s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi hypergraphs: When $c = \Theta(n^{-\alpha})$, our reduction yields different average-case phase diagrams depicting a tradeoff between runtime and $k$ for each fixed $\alpha$. 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 $\frac{\omega k}{3} - C \alpha \binom{k}{2} + O_{k, \alpha}(1)$, where $\frac{\omega}{9} \le C \le 1$ and $\omega$ is the matrix multiplication constant. In the hypergraph case of $s \ge 3$, we show a lower bound at the exponent of $k - \alpha \binom{k}{s} + O_{k, \alpha}(1)$ 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 $c = \Theta(n^{-\alpha})$.