Recognizing Polymatroids Associated with Hypergraphs

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.