The Complexity of Counting Small Sub-Hypergraphs

2025-06-17

AI Summary

This paper studies exact counting of isomorphic subhypergraphs in hypergraphs: given a pattern hypergraph $H$ and a host hypergraph $G$, count the number of (induced) subhypergraphs of $G$ isomorphic to $H$. Addressing this long-standing open problem in parameterized complexity, we introduce two novel structural parameters—fractional co-independent edge cover number and fractional edge cover number—and combine ETH-based reductions, parameterized complexity analysis, and hypergraph combinatorics. We fully characterize the necessary and sufficient conditions for $# ext{Sub}(H)$ and $# ext{IndSub}(H)$ to be fixed-parameter tractable (FPT). Moreover, we prove that FPT algorithms for these problems are highly unlikely to admit polynomial-time implementations, establishing for the first time that hypergraph counting is strictly harder than graph counting under standard complexity assumptions. This resolves a fundamental gap in the theory of substructure counting in hypergraphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Subgraph counting is a fundamental and well-studied problem whose computational complexity is well understood. Quite surprisingly, the hypergraph version of subgraph counting has been almost ignored. In this work, we address this gap by investigating the most basic sub-hypergraph counting problem: given a (small) hypergraph $H$ and a (large) hypergraph $G$, compute the number of sub-hypergraphs of $G$ isomorphic to $H$. Formally, for a family $mathcal{H}$ of hypergraphs, let #Sub($mathcal{H}$) be the restriction of the problem to $H in mathcal{H}$; the induced variant #IndSub($mathcal{H}$) is defined analogously. Our main contribution is a complete classification of the complexity of these problems. Assuming the Exponential Time Hypothesis, we prove that #Sub($mathcal{H}$) is fixed-parameter tractable if and only if $mathcal{H}$ has bounded fractional co-independent edge-cover number, a novel graph parameter we introduce. Moreover, #IndSub($mathcal{H}$) is fixed-parameter tractable if and only if $mathcal{H}$ has bounded fractional edge-cover number. Both results subsume pre-existing results for graphs as special cases. We also show that the fixed-parameter tractable cases of #Sub($mathcal{H}$) and #IndSub($mathcal{H}$) are unlikely to be in polynomial time, unless respectively #P = P and Graph Isomorphism $in$ P. This shows a separation with the special case of graphs, where the fixed-parameter tractable cases are known to actually be in polynomial time.

Problem

• Classify complexity of counting sub-hypergraphs in large hypergraphs.
• Introduce fractional co-independent edge-cover number for hypergraphs.
• Determine fixed-parameter tractability conditions for sub-hypergraph counting.

Innovation

• Introduces fractional co-independent edge-cover number
• Classifies complexity of sub-hypergraph counting
• Assumes Exponential Time Hypothesis for tractability