Decomposition algorithms for solving the minimum weight maximal matching problem

Abstract

We investigate the problem of finding a maximal matching that has minimum total weight on a given edgeweighted graph. Although the minimum weight maximal matching problem is NP-hard in general, polynomial time exact or approximation algorithms on several restricted graph classes are given in the literature. In this article, we propose an exact algorithm for solving several variants of the problem on general graphs. In particular, we develop integer programming (IP) formulations for the problem and devise a decomposition algorithm, which is based on a combination of IP techniques and combinatorial matching algorithms. Our computational tests on a large suite of randomly generated graphs show that our decomposition approach significantly improves the solvability of the problem compared to the underlying IP formulation.

Modelling and solving the perfect edge domination problem

“…The formulation takes edge weights {c e ∈ R : e ∈ E} and specializes to PEDP when {c e = 1 : e ∈ E} applies. Inequalities (2) ensure that at least one edge is selected for every neighborhood N [e], e ∈ E, thus guaranteeing that (1) returns an EDS of G. In turn, as required for perfect edge domination, inequalities (3) enforce that any non dominating edge, i.e., any a ∈ E with x a = 0, must be dominated by no more than one of its neighbor vertices. These inequalities thus become redundant when x a = 1 applies.…”

“…On the other hand, polynomial-time algorithms are known for trees [27], block graphs [10], series-parallel graphs [23], bipartite-permutation graphs [24], and co-triangulated graphs [24]. As for exact algorithms for general graphs, references [2,25] investigate Integer Programming (IP) based approaches for finding Minimum Maximal Matchings (MMMs), where a MMM is a restricted type of EDS (see [2,25] for details).…”

Minimum cost b-matching problems with neighborhoods

“…The minimum-cost perfect bipartite b-matching problem has been studied in Cohen et al [10]. Although the minimum cost maximal matching has been addressed by Bodur et al [9], Taşkin and Ekim [35], Tural [37], to the best of our knowledge, the minimum cost maximal b-matching problem with b > 1 has not been studied.…”

Valid Inequalities for the Maximal Matching Polytope

“…Moreover, they propose some valid inequalities for the problem. The same authors propose a decomposition algorithm for the MWMM problem in [13]. More recently, Tural [14] investigates the IP formulation proposed in [3] and shows that the linear programming (LP) relaxation of it always returns an integral solution for trees and therefore is able to solve the MWMM problem in trees.…”