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Transversals of subtree hypergraphs and the source location problem in hypergraphs.

Heuvel van den, J. and Johnson, M. (2004) 'Transversals of subtree hypergraphs and the source location problem in hypergraphs.', Technical Report. London School of Economics, London.


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 $W\subseteq V$ 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~$S$ of~$V$ such that for any $v\in V$ 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.

Item Type:Monograph (Technical Report)
Additional Information:CDAM Research Reports Series ; LSE-CDAM-2004-10
Keywords:Graphs, Hypergraphs, Source location, Algorithms.
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Record Created:24 Apr 2008
Last Modified:18 Oct 2010 14:46

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