Matchings under distance constraints II
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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.
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