Matchings under distance constraints I.

Abstract

This paper introduces the d-distance matching problem, in which we are given a bipartite graph G=(S,T;E) with S={s_1,dots ,s_n}, a weight function on the edges and an integer din mathbb Z_+. The goal is to find a maximum-weight subset Msubseteq E of the edges satisfying the following two conditions: (i) the degree of every node of S is at most one in M, (ii) if s_it,s_jtin M, then |j-i|ge d. This question arises naturally, for example, in various scheduling problems. We show that the problem is NP-complete in general and admits a simple 3-approximation. We give an FPT algorithm parameterized by d and also show that the case when the size of T is constant can be solved in polynomial time. From an approximability point of view, we show that the integrality gap of the natural integer programming model is at most 2-frac{1}{2d-1}, and give an LP-based approximation algorithm for the weighted case with the same guarantee. A combinatorial (2-frac{1}{d})-approximation algorithm is also presented. Several greedy approaches are considered, and a local search algorithm is described that achieves an approximation ratio of 3/2+epsilon for any constant epsilon >0 in the unweighted case. 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.