Transversals of subtree hypergraphs and the source location problem in digraphs

Heuvel van den, Jan; Johnson, M.

Abstract

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.

Citation

Heuvel van den, J., & Johnson, M. (2008). Transversals of subtree hypergraphs and the source location problem in digraphs. Networks, 51(2), 113-119. https://doi.org/10.1002/net.20206

• Journal Article Type: Article
• Publication Date: Mar 1, 2008
• Deposit Date: Oct 7, 2008
• Publicly Available Date: Dec 11, 2015
• Journal: Networks
• Print ISSN: 0028-3045
• Electronic ISSN: 1097-0037
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 51
• Issue: 2
• Pages: 113-119
• DOI: https://doi.org/10.1002/net.20206
• Keywords: Graphs, Hypergraphs, Source location, Algorithms.
• Public URL: https://durham-repository.worktribe.com/output/1548593