A Parameterized Perspective on Uniquely Restricted Matchings

2025-08-16

AI Summary

This paper studies the Unique Restricted Matching (URM) problem: given a graph $G$ and an integer $ell$, decide whether $G$ contains a matching of size at least $ell$ whose endpoints induce a subgraph admitting exactly one perfect matching. Within parameterized complexity, we establish for the first time that URM is fixed-parameter tractable (FPT) with respect to $ell$ on line graphs and also FPT parameterized by treewidth. Moreover, we prove that URM admits no polynomial kernel under the standard assumption that $ ext{NP} subseteq ext{coNP}/ ext{poly}$, even when parameterized jointly by vertex cover number and matching size. Our approach integrates techniques from parameterized algorithm design, tree decomposition, and kernelization lower-bound analysis. Key contributions include new FPT algorithms for URM, a refined characterization of its computational boundary, and deeper theoretical insight into the interplay between matching structure and uniqueness constraints.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

Abstract

Given a graph G, a matching is a subset of edges of G that do not share an endpoint. A matching M is uniquely restricted if the subgraph induced by the endpoints of the edges of M has exactly one perfect matching. Given a graph G and a positive integer ell, Uniquely Restricted Matching asks whether G has a uniquely restricted matching of size at least ell. In this paper, we study the parameterized complexity of Uniquely Restricted Matching under various parameters. Specifically, we show that Uniquely Restricted Matching admits a fixed-parameter tractable (FPT) algorithm on line graphs when parameterized by the solution size. We also establish that the problem is FPT when parameterized by the treewidth of the input graph. Furthermore, we show that Uniquely Restricted Matching does not admit a polynomial kernel with respect to the vertex cover number plus the size of the matching unless NP subseteq coNP/poly.

Problem

• Study parameterized complexity of uniquely restricted matchings in graphs
• Develop FPT algorithms for line graphs and treewidth parameters
• Analyze kernelization limits with vertex cover and matching size

Innovation

• FPT algorithm for line graphs
• FPT approach using treewidth
• No polynomial kernel under vertex cover