The distance matching problem

Abstract

This paper introduces the d-distance matching problem, in which we are given a bipartite graph (formula presented), a weight function on the edges and an integer (formula presented). The goal is to find a maximum weight subset (formula presented) of the edges satisfying the following two conditions: i) the degree of every node of S is at most one in M, ii) if (formula presented), then (formula presented). The question arises naturally, for example, in various scheduling problems. We show that the problem is NP-complete in general and give an FPT algorithm parameterized by d. We also settle the case when the size of T is constant. From an approximability point of view, we consider a local search algorithm that achieves an approximation ratio of (formula presented) for any constant (formula presented) in the unweighted case. We show that the integrality gap of the natural integer programming model is at most (formula presented), and give an LP-based approximation algorithm for the weighted case with the same guarantee. We also present a combinatorial (formula presented)-approximation algorithm. The novel approaches used in the analysis of the integrality gap and the approximation ratio of locally optimal solutions might be of independent combinatorial interest.