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Transversals of subtree hypergraphs and the source location problem in digraphs

van den Heuvel, Jan ORCID: 0000-0003-0897-9148 and Johnson, Matthew (2004) Transversals of subtree hypergraphs and the source location problem in digraphs. CDAM research report series (CDAM-2004-10). Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science, London, UK.

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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, NP-hard. In this paper, we show that if it is possible to decide if a set of vertices W µ V is a transversal in time S(n) ( where n = |V | ), then it is possible to find a minimum size transversal in O(n3 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 2 V \ K there are k arc-disjoint paths that each join a vertex of K to v and l arc-disjoint 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.

Item Type: Monograph (Report)
Official URL: http://www.cdam.lse.ac.uk
Additional Information: © 2004 the authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Nov 2008 10:42
Last Modified: 19 Sep 2021 23:21
URI: http://eprints.lse.ac.uk/id/eprint/13347

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