A hierarchy of edge-weight symmetries in perfect matchings

2026-04-21

AI Summary

This study investigates whether fixing a single edge suffices to eliminate extremal perfect matchings when the minimum and maximum perfect matching weights differ. By constructing a hierarchical framework of edge-weight symmetries—encompassing node-induced weights, even-cycle symmetry, equiweight perfect matchings, and edge extremality—and integrating block decomposition with tight cut decomposition, the work rigorously characterizes the equivalence and implication relations within this hierarchy for both bipartite and general graphs. The main contributions include proving the full equivalence of this hierarchy in bipartite graphs and, for general graphs, presenting a counterexample demonstrating that fixing one edge is insufficient to exclude all extremal perfect matchings, thereby confirming the tightness of the $2(\ell-1)$ bound (with $\ell=2$) in parameterized algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Motivated by the exact weight perfect matching problem and recent parameterized algorithms for finding an $\ell$-th smallest perfect matching, we study structural properties of edge-weight symmetries in graphs. Recent work by El Maalouly et al. (ESA 2025) showed that excluding all perfect matchings whose weight is at most the $(\ell - 1)$-th smallest possible value in the graph requires fixing at most $2(\ell-1)$ edges in non-bipartite graphs and at most $\ell-1$ edges in bipartite graphs. A natural open question is whether fixing a single edge is always sufficient to shift the extreme (minimum or maximum) weight of a perfect matching when the global minimum and maximum weights differ. To address this, we define and analyze a hierarchy of progressively weaker edge-weight properties: node-induced weights, even walk and cycle symmetries, perfect matching equality, and the edge min-max property. We derive a basic hierarchy among these conditions and show that they become equivalent in bipartite graphs. For general graphs, we provide tight structural characterizations, based on block and tight cut decompositions, under which even cycle symmetry and perfect matching equality force node-induced weights. Finally, we resolve the motivating open question in the negative by constructing a matching-covered non-bipartite graph that satisfies the edge min-max property (every edge is contained in a minimum-weight perfect matching and a maximum-weight one) but violates perfect matching equality (all perfect matchings have the same weight). This counterexample shows that a single edge is not always sufficient to eliminate all minimum-weight or maximum-weight perfect matchings, thereby proving the tightness of the $2(\ell-1)$ bound for $\ell=2$. We also discuss extensions of this framework to $b$-factors and arborescences.

Problem

• perfect matching
• edge-weight symmetry
• min-max property
• matching-covered graph
• extreme weight

Innovation

• edge-weight symmetry
• perfect matching
• min-max property
• matching-covered graph
• tight bound