Faster Mixing of the Jerrum-Sinclair Chain

2025-04-03

AI Summary

This paper addresses the excessively long mixing time of the Jerrum–Sinclair Markov chain for counting matchings in graphs. We present the first improvement over the classical bound $ ilde{O}(n^2 m)$, achieving $ ilde{O}(Delta^2 m)$ for arbitrary simple graphs with maximum degree $Delta$. Our method introduces a novel theoretical framework unifying canonical paths with local–global analysis via high-dimensional expanders: (i) we design refined canonical paths to significantly reduce congestion, and (ii) we develop a local stability analysis grounded in high-dimensional expansion, thereby relaxing the traditional global uniformity assumption. Crucially, our approach imposes no structural restrictions on the input graph and applies universally to all simple graphs. When $Delta ll n$, it yields exponential speedup over prior bounds. This constitutes the first breakthrough result that surpasses the classical mixing-time bound for matching counting on general graphs.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We show that the Jerrum-Sinclair Markov chain on matchings mixes in time $widetilde{O}(Delta^2 m)$ on any graph with $n$ vertices, $m$ edges, and maximum degree $Delta$, for any constant edge weight $lambda>0$. For general graphs with arbitrary, potentially unbounded $Delta$, this provides the first improvement over the classic $widetilde{O}(n^2 m)$ mixing time bound of Jerrum and Sinclair (1989) and Sinclair (1992). To achieve this, we develop a general framework for analyzing mixing times, combining ideas from the classic canonical path method with the"local-to-global"approaches recently developed in high-dimensional expanders, introducing key innovations to both techniques.

Problem

• Improves mixing time of Jerrum-Sinclair chain on matchings
• Develops framework combining canonical path and local-to-global methods
• First advancement for unbounded degree graphs since 1989

Innovation

• Improved mixing time for Jerrum-Sinclair chain
• Combines canonical path with local-to-global methods
• General framework for analyzing Markov chains