Modeling Hypergraphs by Graphs with the Same Mincut Properties

An elegant and general way to apply graph partitioning algorithms to hypergraphs would be to model hypergraphs by graphs and apply the graph algorithms to these models Of course such models have to simulate the given hypergraphs with respect to their cut properties An edge weighted graph V E is a cut model for an edge weighted hypergraph V H if the weight of the edges cut by any bipartition of V in the graph is the same as the weight of the hyperedges cut by the same bipartition in the hypergraph We show that there is no cut model in general Next we examine whether the addition of dummy vertices helps An edge weighted graph V D E is a mincut model for an edge weighted hypergraph V H if the weight of the hyperedges cut by a bipartition of the hypergraphs vertices is the same as the weight of a minimum cut separating the two parts in the graph We construct such models using positive and negative weights On the other hand we show that there is no mincut model in general if only positive weights are allowed This work was done while the authors were with THE LEONARDO FIBONACCI INSTITUTE for the Foundations of Computer Science Theoretische Informatik Eidgen ossische Technische Hochschule Z urich CH Z urich e mail ih ler inf ethz ch Fachbereich Mathematik Technische Universit at Berlin Stra e des Juni W Berlin Ger many e mail dorothea combi math tu berlin de Institut f ur Informatik Fachbereich Mathematik Freie Universit at Berlin Arnimallee W Berlin Germany e mail wagner tcs fu berlin de This work was partly supported by the ESPRIT Basic Research Action Projekt ALCOM I