Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs

2025-11-26

AI Summary

This paper addresses the problem of determining the existence of a perfect matching in bipartite graphs of genus $O(log n)$. For this graph class, we construct—*for the first time*—a deterministic, polynomially bounded edge-weight function computable in logarithmic space ($mathsf{L}$) that isolates a unique minimum-weight perfect matching. Our method integrates graph embedding theory, algebraic techniques (including determinant-based isolation), and logspace computation primitives to circumvent the failure of classical isolation lemmas on high-genus surfaces. The main contribution is establishing that this problem lies in $mathsf{SPL}$—the class of problems solvable by deterministic logspace machines with access to a symmetric logarithmic-space oracle—thereby resolving its complexity from previously unknown or higher classes (e.g., $mathsf{NL}$ or beyond) to deterministic logspace-reducible isolation. This constitutes a key advance in low-complexity graph matching theory and provides a foundational step toward $mathsf{NC}$- or $mathsf{L}$-time matching algorithms for embedded bipartite graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

We show that given an embedding of an $O(log n)$ genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists. As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for $O(log n)$ genus bipartite graphs.

Problem

• Isolating perfect matchings in logarithmic genus bipartite graphs
• Constructing logspace edge weights for unique matchings
• Solving perfect matching decision problem in SPL complexity

Innovation

• Logspace edge weight construction for uniqueness
• Isolates minimum weight perfect matching deterministically
• Applies to O(log n) genus bipartite graphs