On Tree-Constrained Matchings and Generalizations

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 = 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). ...

• Type: Journal
• Year: 2011
• Where: ICALP
• Authors: Stefan Canzar, Khaled M. Elbassioni, Gunnar W. Klau, Julián Mestre
• Keywords: Approximation Algorithm, E1 T2, ICALP 2011, Natural Generalization, Programming Languages