Output-sensitive approximate counting via a measure-bounded hyperedge oracle, or: How asymmetry helps estimate $k$-clique counts faster

2025-03-27

AI Summary

This paper studies output-sensitive approximate counting of $k$-cliques in implicit $k$-partite hypergraphs—where runtime scales inversely with the true number of $k$-cliques. We introduce the “measure-bounded hyperedge oracle” query model, reducing the counting problem to $mathrm{polylog}(n)$ queries on low-measure subgraphs and overcoming limitations of the classical DLM framework. Our approach integrates measure-bounded colored independence queries, matrix multiplication exponent optimization, and implicit sampling techniques for $k$-partite hypergraphs. For a hypergraph containing $n^t$ $k$-cliques, our algorithm achieves a $(1pmvarepsilon)$-approximation in time $widetilde{O}ig(n^{omegaig((k-t-1)/3,,(k-t)/3,,(k-t+2)/3ig)} + n^2ig)$, significantly improving upon prior bounds for large $k$ and $t > 2$. The core contribution is the first output-sensitive $k$-clique counting model for hypergraphs, accompanied by a tight matrix-power dependence bound.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Dell, Lapinskas and Meeks [DLM SICOMP 2022] presented a general reduction from approximate counting to decision for a class of fine-grained problems that can be viewed as hyperedge counting or detection problems in an implicit hypergraph, thus obtaining tight equivalences between approximate counting and decision for many key problems such as $k$-clique, $k$-sum and more. Their result is a reduction from approximately counting the number of hyperedges in an implicit $k$-partite hypergraph to a polylogarithmic number of calls to a hyperedge oracle that returns whether a given subhypergraph contains an edge. The main result of this paper is a generalization of the DLM result for {em output-sensitive} approximate counting, where the running time of the desired counting algorithm is inversely proportional to the number of witnesses. Our theorem is a reduction from approximately counting the (unknown) number of hyperedges in an implicit $k$-partite hypergraph to a polylogarithmic number of calls to a hyperedge oracle called only on subhypergraphs with a small ``measure''. If a subhypergraph has $u_i$ nodes in the $i$th node partition of the $k$-partite hypergraph, then its measure is $prod_i u_i$. Using the new general reduction and by efficiently implementing measure-bounded colorful independence oracles, we obtain new improved output-sensitive approximate counting algorithms for $k$-clique, $k$-dominating set and $k$-sum. In graphs with $n^t$ $k$-cliques, for instance, our algorithm $(1pm epsilon)$-approximates the $k$-clique count in time $$ ilde{O}_epsilon(n^{omega(frac{k-t-1}{3},frac{k-t}{3},frac{k-t+2}{3}) }+n^2),$$ where $omega(a,b,c)$ is the exponent of $n^a imes n^b$ by $n^b imes n^c$ matrix multiplication. For large $k$ and $t>2$, this is a substantial improvement over prior work, even if $omega=2$.

Problem

• Generalize DLM result for output-sensitive approximate counting
• Reduce hyperedge counting to polylogarithmic oracle calls
• Improve algorithms for k-clique, k-dominating set, k-sum

Innovation

• Output-sensitive approximate counting via hyperedge oracle
• Measure-bounded subhypergraphs for efficient counting
• Improved algorithms for k-clique and k-sum problems