Publication details for Professor Matthew JohnsonHeuvel van den, Jan & Johnson, M. (2008). Transversals of subtree hypergraphs and the source location problem in digraphs. Networks 51(2): 113-119.
- Publication type: Journal Article
- ISSN/ISBN: 0028-3045, 1097-0037
- DOI: 10.1002/net.20206
- Keywords: Graphs, Hypergraphs, Source location, Algorithms.
- Further publication details on publisher web site
- Durham Research Online (DRO) - may include full text
Author(s) from Durham
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.
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