Applications of the Dulmage-mendelsohn Decomposition and Network Flow to Graph Bisection Improvement

In this paper, we consider the use of the Dulmage-Mendelsohn decomposition and network ow on bipartite graphs to improve a graph bisection partition. Given a graph partition S; B; W ] with a vertex separator S and two disconnected components B and W , diierent strategies are considered based on the Dulmage-Mendelsohn decomposition to reduce the separator size jSj and/or the imbalance between B and W. For the case when the vertices are weighted, we relate this with the bipartite network ow problem. A further enhancement is made on partition improvement by generalizing the bipartite network to solving a general network ow problem. We demonstrate the utility of these improvement techniques on a set of sparse test matrices, where we nd top level separators and nested dissection and multisection orderings.