Minimum cost b-matching problems with neighborhoods

Abstract

In this paper, we deal with minimum cost b-matching problems on graphs where the nodes are assumed to belong to non-necessarily convex regions called neighborhoods, and the costs are given by the distances between points of the neighborhoods. The goal in the proposed problems is twofold: (i) finding a b-matching in the graph and (ii) determining a point in each neighborhood to be the connection point among the edges defining the b-matching. Different variants of the minimum cost b-matching problem are considered depending on the criteria to match neighborhoods: perfect, maximum cardinality, maximal and the a-b-matching problems. The theoretical complexity of solving each one of these problems is analyzed. Different mixed integer non-linear programming formulations are proposed for each one of the considered problems and then reformulated as Second Order Cone formulations. An extensive computational experience shows the efficiency of the proposed formulations to solve the problems under study.