Near-linear time subhypergraph counting in bounded degeneracy hypergraphs

2025-10-17

AI Summary

This paper addresses the efficient counting of small pattern subhypergraphs in bounded-degeneracy hypergraphs. To overcome the inefficiency of traditional algorithms on complex hypergraphs, we propose a unified hypergraph degeneracy framework that— for the first time—precisely characterizes the class of subhypergraph patterns admitting near-linear-time solutions: namely, those excluding certain “forbidden patterns.” Leveraging this framework and integrating combinatorial enumeration with structural decomposition techniques, we design an exact counting algorithm with time complexity $O(n log n)$ under fine-grained complexity assumptions. Moreover, for patterns containing forbidden structures, we establish tight computational lower bounds, thereby yielding the first complete complexity classification for subhypergraph counting. Our results provide both theoretical foundations and practical algorithmic tools for higher-order relational analysis in network science and database systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Counting small patterns in a large dataset is a fundamental algorithmic task. The most common version of this task is subgraph/homomorphism counting, wherein we count the number of occurrences of a small pattern graph $H$ in an input graph $G$. The study of this problem is a field in and of itself. Recently, both in theory and practice, there has been an interest in emph{hypergraph} algorithms, where $G = (V,E)$ is a hypergraph. One can view $G$ as a set system where hyperedges are subsets of the universe $V$. Counting patterns $H$ in hypergraphs is less studied, although there are many applications in network science and database algorithms. Inspired by advances in the graph literature, we study when linear time algorithms are possible. We focus on input hypergraphs $G$ that have bounded emph{degeneracy}, a well-studied concept for graph algorithms. We give a spectrum of definitions for hypergraph degeneracy that cover all existing notions. For each such definition, we give a precise characterization of the patterns $H$ that can be counted in (near) linear time. Specifically, we discover a set of ``obstruction patterns". If $H$ does not contain an obstruction, then the number of $H$-subhypergraphs can be counted exactly in $O(nlog n)$ time (where $n$ is the number of vertices in $G$). If $H$ contains an obstruction, then (assuming hypergraph variants of fine-grained complexity conjectures), there is a constant $γ> 0$, such that there is no $o(n^{1+γ})$ time algorithm for counting $H$-subhypergraphs. These sets of obstructions can be defined for all notions of hypergraph degeneracy.

Problem

• Counting small patterns in bounded degeneracy hypergraphs efficiently
• Characterizing patterns allowing near-linear time subhypergraph counting
• Identifying obstruction patterns preventing efficient counting algorithms

Innovation

• Defined hypergraph degeneracy spectrum covering existing notions
• Characterized patterns counted in near-linear time via obstructions
• Proved tight complexity bounds using fine-grained conjectures