Centre for Discrete and Applicable Mathematics 

CDAM Research Report, LSECDAM200410May 2004 
Jan van den Heuvel and Matthew Johnson
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. To find a minimum size transversal for a subtree hypergraph is, in general, NPhard. In this paper, we show that if it is possible to decide if a set of vertices W in V is a transversal in time S(n) (where n = V), then it is possible to find a minimum size transversal in O(n^{3}S(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 K of V such that for any v in V\K there are k arcdisjoint paths that each join a vertex of K to v and l arcdisjoint paths that each join v to K. 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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(This paper supersedes the earlier paper A Polynomial Algorithm for the Source Location Problem in Digraphs, which is obsolete.)
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