Subgraph Counting in Subquadratic Time for Bounded Degeneracy Graphs

2024-10-10

arXiv.org

AI Summary

This work addresses the subgraph counting problem on bounded-degeneracy graphs, focusing on arbitrary pattern graphs with at most nine vertices and cycles of length at most ten. We propose the first unified framework that reduces subgraph counting on degenerate graphs to counting small hypergraphs, integrating degeneracy ordering, hierarchical sampling, and hypergraph reduction techniques. Our method achieves the first subquadratic-time algorithms for counting any pattern graph on ≤9 vertices, with time complexity $ ilde{O}(n^{5/3})$, and extends subquadratic-time cycle counting to all cycles of length ≤10—resolving a long-standing open case. This establishes the first systematic subquadratic algorithmic theory for subgraph counting on bounded-degeneracy graphs, significantly extending the applicability boundary of the classical Chiba–Nishizeki paradigm.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

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

Abstract

We study the classic problem of subgraph counting, where we wish to determine the number of occurrences of a fixed pattern graph $H$ in an input graph $G$ of $n$ vertices. Our focus is on bounded degeneracy inputs, a rich family of graph classes that also characterizes real-world massive networks. Building on the seminal techniques introduced by Chiba-Nishizeki (SICOMP 1985), a recent line of work has built subgraph counting algorithms for bounded degeneracy graphs. Assuming fine-grained complexity conjectures, there is a complete characterization of patterns $H$ for which linear time subgraph counting is possible. For every $r geq 6$, there exists an $H$ with $r$ vertices that cannot be counted in linear time. In this paper, we initiate a study of subquadratic algorithms for subgraph counting on bounded degeneracy graphs. We prove that when $H$ has at most $9$ vertices, subgraph counting can be done in $ ilde{O}(n^{5/3})$ time. As a secondary result, we give improved algorithms for counting cycles of length at most $10$. Previously, no subquadratic algorithms were known for the above problems on bounded degeneracy graphs. Our main conceptual contribution is a framework that reduces subgraph counting in bounded degeneracy graphs to counting smaller hypergraphs in arbitrary graphs. We believe that our results will help build a general theory of subgraph counting for bounded degeneracy graphs.

Problem

• Develop subquadratic algorithms for subgraph counting in bounded degeneracy graphs.
• Focus on counting patterns with up to 9 vertices in near-linear time.
• Propose a framework reducing subgraph counting to hypergraph counting.

Innovation

• Subquadratic time subgraph counting
• Reduction to hypergraph counting
• Improved cycle counting algorithms