The Fine-Grained Complexity of Counting Hypergraph Motifs

2026-07-06

AI Summary

This study investigates the fine-grained complexity of exactly counting three-hyperedge motifs in hypergraphs that realize arbitrary Venn diagram intersection patterns, parameterized by hypergraph rank. Leveraging the Triangle Detection and Hyperclique hypotheses within the framework of parameterized complexity theory and combinatorial analysis, the work provides the first complete characterization of the complexity landscape for all such patterns. It establishes that a fixed-parameter near-linear time algorithm exists if and only if the Venn diagram corresponds to a degenerate case—specifically, when one hyperedge is entirely contained within another. For all non-degenerate patterns, the problem provably requires nearly quadratic time. This result yields a comprehensive and precise map of the computational complexity of motif counting under rank parameterization.

Abstract

Introduced by Lee, Ko, and Shin (VLDB 2020), a hypergraph motif is a connected subhypergraph consisting of three hyperedges whose intersections satisfy a prescribed pattern. Such patterns are represented by Venn diagrams $\mathcal{V}\in\{0,1\}^7$, indicating which of the seven regions determined by three sets must be empty or non-empty. Lee et al. designed and implemented exact and approximate algorithms for counting, in a hypergraph $G$, the motifs specified by $\mathcal{V}$; their algorithms run in worst-case cubic time in the number of hyperedges of $G$. This cubic worst case can occur even for hypergraphs of bounded rank, and already for $2$-uniform hypergraphs, that is, for simple graphs. In this work, we give a complete fine-grained picture of the parameterised complexity of exact hypergraph motif counting with respect to the rank of the input hypergraph. We use $\tilde{O}$ to hide polylogarithmic factors in the input size. First, we show that every Venn diagram $\mathcal{V}$ admits an exact counting algorithm running in FPT-near-quadratic time, \[ f(\mathsf{rank}(G))\cdot \tilde{O}(|E(G)|^2), \] for some computable function $f$. Second, we precisely characterise when this can be improved to FPT-near-linear time. We prove that such an algorithm exists exactly for the degenerate Venn diagrams, namely those that force one of the three hyperedges to be fully contained in another. For all non-degenerate Venn diagrams, we show that no FPT-near-linear-time algorithm exists unless either the Triangle Hypothesis or the Hyperclique Hypothesis fails. Exact hypergraph motif counting is thus always fixed-parameter near-quadratic in the rank, and the degenerate Venn diagrams are precisely the cases admitting fixed-parameter near-linear time.

Problem

• hypergraph motif
• fine-grained complexity
• parameterized complexity
• Venn diagram
• motif counting

Innovation

• fine-grained complexity
• hypergraph motif counting
• fixed-parameter tractability
• degenerate Venn diagrams
• Triangle Hypothesis