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

By J. Heuvel van den and M. Johnson

Abstract

A hypergraph $H=(V,E)$ is a subtree hypergraph if there is a tree~$T$\ud on~$V$ such that each hyperedge of~$E$ induces a subtree of~$T$.\ud Since the number of edges of a subtree hypergraph can be exponential in\ud $n=|V|$, one can not always expect to be able to find a minimum size\ud transversal in time polynomial in~$n$. In this paper, we show that if it is\ud possible to decide if a set of vertices $W\subseteq V$ is a transversal in\ud time~$S(n)$ (\,where $n=|V|$\,), then it is possible to find a minimum size\ud transversal in~$O(n^3\,S(n))$.\ud \ud This result provides a polynomial algorithm for the Source Location\ud Problem\,: a set of $(k,l)$-sources for a digraph $D=(V,A)$ is a subset~$S$\ud of~$V$ such that for any $v\in V$ there are~$k$ arc-disjoint paths that\ud each join a vertex of~$S$ to~$v$ and~$l$ arc-disjoint paths that each\ud join~$v$ to~$S$. The Source Location Problem is to find a minimum size set\ud of $(k,l)$-sources. We show that this is a case of finding a transversal of\ud a subtree hypergraph, and that in this case~$S(n)$ is polynomial

Topics: Graphs, Hypergraphs, Source location, Algorithms.
Publisher: London School of Economics
Year: 2004
OAI identifier: oai:dro.dur.ac.uk.OAI2:620

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