Transversals of subtree hypergraphs and the source location problem in digraphs

A hypergraph H = (V,E) is a subtree hypergraph if there is a tree T on V such that each hyperedge of E induces a subtree of T. Since the number of edges of a subtree hypergraph can be exponential in n = |V|, one can not always expect to be able to find a minimum size transversal in time polynomial in n. In this paper, we show that if it is possible to decide if a set of vertices W V is a transversal in time S(n) (where n = |V|), then it is possible to find a minimum size transversal in O(n3S(n)). This result provides a polynomial algorithm for the Source Location Problem: a set of (k,l)-sources for a digraph D = (V,A) is a subset S of V such that for any v V there are k arc-disjoint paths that each join a vertex of S to v and l arc-disjoint paths that each join v to S. The Source Location Problem is to find a minimum size set of (k,l)-sources. We show that this is a case of finding a transversal of a subtree hypergraph, and that in this case S(n) is polynomial.