On tree-constrained matchings and generalizations

Stefan Canzar, Khaled Elbassioni, Gunnar W. Klau, Julián Mestre

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Scopus citations

Abstract

We consider the following Tree-Constrained Bipartite Matching problem: Given two rooted trees T 1 = (V 1,E 1), T 2 = (V 2,E 2) and a weight function w: V 1x V 2 → ℝ+, find a maximum weight matching 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 -hard and thus, unless , 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.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings
Pages98-109
Number of pages12
EditionPART 1
DOIs
StatePublished - 2011
Event38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland
Duration: Jul 4 2011Jul 8 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume6755 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other38th International Colloquium on Automata, Languages and Programming, ICALP 2011
Country/TerritorySwitzerland
CityZurich
Period7/4/117/8/11

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

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