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On Tree-Constrained Matchings and Generalizations (2011)
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BibTeX
@MISC{Canzar11ontree-constrained,
author = {Stefan Canzar and Khaled Elbassioni and Gunnar W. Klau and Julián Mestre},
title = {On Tree-Constrained Matchings and Generalizations },
year = {2011}
}
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Abstract
We consider the following Tree-Constrained Bipartite Matching problem: Given two rooted trees T1 = (V1, E1), T2 = (V2, E2) and a weight function w: V1 × V2 ↦ → R+, find a maximum weight matching M between nodes of the two trees, 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 = N P, 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.
Keyphrases
tree-constrained matchings matched node local ratio technique v1 v2 second part computational analysis rooted tree 2-approximation algorithm polynomial time k-dimensional matching generalization maximum weight incomparable element previous claim inapproximability result matching integrality gap example following tree-constrained bipartite matching problem maximum number natural lp-relaxation basic feasible solution classical bipartite weight function natural generalization careful use integrality gap live cell video data
