# The Average-Case Complexity of Counting Cliques in Erdos-Renyi Hypergraphs

## TL;DR

This paper analyzes the average-case complexity of counting k-cliques in Erdős-Rényi hypergraphs, showing it aligns with worst-case complexity in dense cases and providing faster algorithms for sparse cases, with implications for phase transitions.

## Contribution

It establishes a worst-case-to-average-case complexity reduction for clique counting in Erdős-Rényi hypergraphs and introduces algorithms exploiting hypergraph sparsity.

## Key findings

- Counting k-cliques in dense hypergraphs matches worst-case complexity up to polylog factors.
- New algorithms outperform worst-case bounds in sparse hypergraphs.
- Identifies a phase transition in complexity related to the Erdős-Rényi k-clique percolation threshold.

## Abstract

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 graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized 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 graphs and hypergraphs: When $c = \Theta(n^{-\alpha})$, we give several algorithms exploiting the sparsity of $G(n, c, s)$ that are faster than the best known worst-case algorithms. Complementing this, based on a fine-grained worst-case assumption, our results imply a different average-case phase diagram for each fixed $\alpha$ depicting a tradeoff between a runtime lower bound and $k$. Surprisingly, in the hypergraph case ($s \ge 3$), these lower bounds are tight against our algorithms exactly when $c$ is above the Erd\H{o}s-R\'{e}nyi $k$-clique percolation threshold. This is the first worst-case-to-average-case hardness reduction for a problem on Erd\H{o}s-R\'{e}nyi hypergraphs that we are aware of. We also give a variant of our result for computing the parity of the $k$-clique count that tolerates higher error probability.