Matchings under distance constraints II.

This paper introduces the d -distance b -matching problem , in which we are given a bipartite graph G=(S,T;E) with S={s_1,… ,s_n} , a weight function on the edges, an integer d∈ℤ_+ and a degree bound function b:S∪ T→ℤ_+ . The goal is to find a maximum-weight subset M⊆ E of the edges satisfying the following two conditions: (1) the degree of each node v∈ S∪ T is at most b ( v ) in M , (2) if s_it,s_jt∈ M , then |i-j|≥ d . In the cyclic version of the problem, the nodes in S are considered to be in cyclic order. We get back the (cyclic) d -distance matching problem when b(s) = 1 for s∈ S and b(t) = ∞ for t∈ T . We prove that the d -distance matching problem is APX-hard, even in the unweighted case. We show that 2-1/d is a tight upper bound on the integrality gap of the natural integer programming model for the cyclic d -distance b -matching problem provided that (2d-1) divides the size of S . For the non-cyclic case, the integrality gap is shown to be at most (2-2/d) . The proofs give approximation algorithms with guarantees matching these bounds, and also improve the best known algorithms for the (cyclic) d -distance matching problem. In a related problem, our goal is to find a permutation of S maximizing the weight of the optimal d -distance b -matching. This problem can be solved in polynomial time for the (cyclic) d -distance matching problem — even though the (cyclic) d -distance matching problem itself is NP-hard and also hard to approximate arbitrarily. For (cyclic) d -distance b -matchings, however, we prove that finding the best permutation is NP-hard, even if b≡ 2 or d=2 , and we give e -approximation algorithms.