Minmax-cut graph partitioning problems

Two partitioning problems on graphs are considered. In the first problem the nodes of a directed graph are partitioned into sets of sizes within prescribed ranges. If in(V_i) is the sum of the weights on the incoming edges to set V_i in the partition, the goal is to minimize max_i {in( V_i)}. It is shown that the problem is NP-hard if the maximum set size is at least three or there is a constant number of sets of the same size. For the case where n and m are the number of nodes and the number of edges of the input graph, respectively, an O(m√n) time algorithm is obtained when the maximum set size is two. The same problem is then considered on undirected graphs. It is shown that this partitioning problem is NP-hard for partitioning into equal-size sets, but polynomial-time algorithms are obtained when the maximum set size is a constant k. Applications of the problems are in layout, built-in self-test (BIST), and high-level synthesis