Partitioning a Multi-weighted Graph to Connected Subgraphs of Almost Uniform Size

Assume that each vertex of a graph G is assigned a constant number q of nonnegative integer weights, and that q pairs of nonnegative integers li and ui, 1 ≤ i ≤ q, are given. One wishes to partition G into connected components by deleting edges from G so that the total i-th weights of all vertices in each component is at least li and at most ui for each index i, 1 ≤ i ≤ q. The problem of finding such a “uniform” partition is NP-hard for series-parallel graphs, and is strongly NP-hard for general graphs even for q = 1. In this paper we show that the problem and many variants can be solved in pseudo-polynomial time for series-parallel graphs and partial k-trees, that is, graphs with bounded tree-width. key words: algorithm, choice partition, lower bound, maximum partition problem, minimum partition problem, multi-weighted graph, partial k-tree, series-parallel graph, uniform partition, upper bound