On Tree-Constrained Matchings and Generalizations

We consider the following Tree-Constrained Bipartite Matching problem: Given a bipartite graph G=(V1,V2,E) with edge weights (Formula presented.), a rooted tree T1 on the set V1 and a rooted tree T2 on the set V1, find a maximum weight matching (Formula presented.)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 (Formula presented.)-hard and thus, unless (Formula presented.), 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).