The biobjective minimum-cost perfect matching problem and Chinese postman problem

Abstract

In this paper, we address the biobjective versions of the perfect matching problem (PMP) and the Chinese postman problem (CPP). Both problems are solved by means of integer formulations or separating blossom inequalities, exploiting the PMP relationship with the CPP. In both cases, we first find the set of supported nondominated solutions and then we use them to obtain the nonsupported ones. The set of supported nondominated solutions are obtained solving scalarized integer formulations. To obtain the sets of nonsupported solutions, we resort to solving lexicographic problems based on adding additional linear constraints to the original problems. For this reason, we also characterize the combinatorial structure of the PMP vertices with one or two additional constraints. We also investigate when it is possible to use the PMP to solve CPP in the biobjective case. We report computational experiments comparing the different approaches and formulations based on different types of graphs with up to 700 nodes.