Polyhedral approach to weighted connected matchings in general graphs

TL;DR

The paper addresses the NP-hard problem of finding maximum-weight connected matchings in general graphs. It develops two polyhedral MILP formulations—a compact extended flow-based formulation and a high-level original-space formulation using strong inequalities from the matching and connected-subgraph polytopes—and pairs them with a sophisticated branch-and-cut scheme. The exponential formulation employs separation procedures for minimal separators, indegree inequalities, and blossom cuts, and is implemented with open-source code. Experimental results on DIMACS MWCS/GMWCS benchmarks show the exponential approach solving the majority of instances and providing a strong baseline for future work, while the extended formulation offers solid performance on easier cases. Overall, the work demonstrates that polyhedral MILP techniques can yield practical, scalable solutions for weighted connected matchings and establishes a reproducible framework for further research.

Abstract

A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to determine an optimal connected matching in an edge-weighted graph, even in the planar bipartite case. We present two mixed integer programming formulations and a sophisticated branch-and-cut scheme to find weighted connected matchings in general graphs. The formulations explore different polyhedra associated to this problem, including strong valid inequalities both from the matching polytope and from the connected subgraph polytope. We conjecture that one attains a tight approximation of the convex hull of connected matchings using our strongest formulation, and report encouraging computational results over DIMACS Implementation Challenge benchmark instances. The source code of the complete implementation is also made available.