Matchings under distance constraints II

This paper introduces the d-distance b-matching problem, in which we are given a bipartite graph $$G=(S,T;E)$$ G = ( S , T ; E ) with $$S=\{s_1,\dots ,s_n\}$$ S = { s 1 , ⋯ , s n } , a weight function on the edges, an integer $$d\in \mathbb {Z}_+$$ d ∈ Z + and a degree bound function $$b:S\cup T\rightarrow \mathbb {Z}_+$$ b : S ∪ T → Z + . The goal is to find a maximum-weight subset $$M\subseteq E$$ M ⊆ E of the edges satisfying the following two conditions: (1) the degree of each node $$v\in S\cup T$$ v ∈ S ∪ T is at most b(v) in M, (2) if $$s_it,s_jt\in M$$ s i t , s j t ∈ M , then $$|i-j|\ge d$$ | 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 $$b(s) = 1$$ b ( s ) = 1 for $$s\in S$$ s ∈ S and $$b(t) = \infty $$ b ( t ) = ∞ for $$t\in T$$ t ∈ T . We prove that the d-distance matching problem is APX-hard, even in the unweighted case. We show that $$2-\frac{1}{d}$$ 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 $$(2d-1)$$ ( 2 d - 1 ) divides the size of S. For the non-cyclic case, the integrality gap is shown to be at most $$(2-\frac{2}{d})$$ ( 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 $$b\equiv 2$$ b ≡ 2 or $$d=2$$ d = 2 , and we give e-approximation algorithms.