Optimal Hypergraph Tree-Realization

Abstract

Consider a hyperstar H and a function ω assigning a non-negative weight to every unordered pair of vertices of H and satisfying the following restriction: for any three vertices u,v,x such that u and v belong to the same set of hyperedges, ω ({u,x}) = ω ({v,x}). We provide an efficient method that finds a tree-realization T of H which has the maximum weight subject to the minimum number of leaves.

We transform the problem to the construction of an optimal degree-constrained spanning arborescence of a non-negatively weighted directed acyclic graph (DAG). The latter problem is a special case of the weighted matroid intersection problem. We propose a faster method based on finding the maximum weighted bipartite matching.