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THE AVERAGE-CASE COMPLEXITY OF COUNTING CLIQUES IN ERDOŐS-REÉNYI HYPERGRAPHS.

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Title: THE AVERAGE-CASE COMPLEXITY OF COUNTING CLIQUES IN ERDOŐS-REÉNYI HYPERGRAPHS.
Authors: BOIX-ADSERAÀ, ENRIC, BRENNAN, MATTHEW, BRESLER, GUY
Source: SIAM Journal on Computing; 2025, Vol. 54 Issue 4, p39-80, 42p
Subject Terms: COMPUTATIONAL complexity, HYPERGRAPHS, SPARSE graphs, COMBINATORICS, TIME complexity, SUBGRAPHS
Abstract: 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 prove the following: (1) 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 randomized ETH, it takes nΩ(k) time to count k-cliques in G(n, c, s) if k and c are constant. (2) Sparse Erdős--Rényi graphs and hypergraphs: When c = θ(n-α), we give several algorithms exploiting the sparsity of G(n, c, s) that are faster than the best known worstcase algorithms. Complementing this, based on a fine-grained worst-case assumption, our reduction implies a different average-case phase diagram for each fixed α depicting a tradeoff between a runtime lower bound and k. Surprisingly, in the hypergraph case (s ≥ 3), these lower bounds are tight against our algorithms exactly when c is above the Erdős--Rényi k-clique percolation threshold. Our reduction yields the first known average-case hardness result on Erdős--Rényi hypergraphs based on worst-case hardness conjectures. We also give a variant of our worst-case to average-case reduction for computing the parity of the k-clique count that requires a milder assumption on the error probability of the blackbox solving the problem on G(n, c, s). [ABSTRACT FROM AUTHOR]
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Abstract: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 prove the following: (1) 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 randomized ETH, it takes n<sup>Ω(k)</sup> time to count k-cliques in G(n, c, s) if k and c are constant. (2) Sparse Erdős--Rényi graphs and hypergraphs: When c = θ(n<sup>-α</sup>), we give several algorithms exploiting the sparsity of G(n, c, s) that are faster than the best known worstcase algorithms. Complementing this, based on a fine-grained worst-case assumption, our reduction implies a different average-case phase diagram for each fixed α depicting a tradeoff between a runtime lower bound and k. Surprisingly, in the hypergraph case (s ≥ 3), these lower bounds are tight against our algorithms exactly when c is above the Erdős--Rényi k-clique percolation threshold. Our reduction yields the first known average-case hardness result on Erdős--Rényi hypergraphs based on worst-case hardness conjectures. We also give a variant of our worst-case to average-case reduction for computing the parity of the k-clique count that requires a milder assumption on the error probability of the blackbox solving the problem on G(n, c, s). [ABSTRACT FROM AUTHOR]
ISSN:00975397
DOI:10.1137/20M1316044