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Recognizing Polymatroids Associated with Hypergraphs

Published online by Cambridge University Press:  12 September 2008

Dirk Vertigan  and
Geoff Whittle
Dirk Vertigan
Affiliation:
Mathematical Institute, Oxford University, England
Geoff Whittle
Affiliation:
Department of Mathematics, University of Tasmania, Hobart, Tasmania 7001, Australia
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Abstract

Two natural classes of polymatroids can be associated with hypergraphs: the so-called Boolean and hypergraphic polymatroids. Boolean polymatroids carry virtually all the structure of hypergraphs; hypergraphic polymatroids generalize graphic matroids. This paper considers algorithmic problems associated with recognizing members of these classes. Let k be a fixed positive integer and assume that the k-polymatroid ρ is presented via a rank oracle. We present an algorithm that determines in polynomial time whether ρ is Boolean, and if it is, finds the hypergraph. We also give an algorithm that decides in polynomial time whether ρ is the hypergraphic polymatroid associated with a given hypergraph. Other structure-theoretic results are also given.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 2 , Issue 4 , December 1993 , pp. 519 - 530
Copyright
Copyright © Cambridge University Press 1993

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