Lower bounds and exact algorithms for the graph partitioning problem using multicommodity flows

In this paper new and generalized lower bounds for the graph partitioning problem are presented. These bounds base on the well known lower bound of embedding a clique into the given graph with minimal congestion which is equivalent to a multicommodity flow problem where each vertex sends a commodity of size one to every other vertex. Our new bounds use arbitrary multicommodity flow instances for the bound calculation, the critical point for the lower bound is the guaranteed cut flow of the instances. Furthermore, a branch&bound procedure basing on these bounds is presented. Finally, upper bounds of the lower bounds are shown which demonstrate the superiority of the presented generalizations; and the new bounds are applied to the Butterfly and Beneš network.