On Tree-Constrained Matchings and Generalizations

Abstract : We consider the following Tree-Constrained Bipartite Matching problem: Given a bipartite graph G=(V1,V2,E) with edge weights w:E↦ℝ+w:E↦R+, a rooted tree T1 on the set V1 and a rooted tree T2 on the set V1, find a maximum weight matching M in G, such that none of the matched nodes is an ancestor of another matched node in either of the trees. This generalization of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We show that the problem is APX-hard and thus, unless =P=NP, disprove a previous claim that it is solvable in polynomial time. Furthermore, we give a 2-approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of 2−o(1). In the second part of the paper, we consider a natural generalization of the problem, where trees are replaced by partially ordered sets (posets). We show that the local ratio technique gives a 2kρ-approximation for the k-dimensional matching generalization of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by ρ. We finally give an almost matching integrality gap example, and an inapproximability result showing that the dependence on ρ is most likely unavoidable.
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Stefan Canzar, Khaled Elbassioni, Gunnar W. Klau, Julián Mestre. On Tree-Constrained Matchings and Generalizations. Algorithmica, Springer Verlag, 2015, 71 (1), ⟨10.1007/s00453-013-9785-0⟩. ⟨hal-01248542⟩

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