Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

Abstract

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u. Such an “almost uniform” partition is called an $(l,u)$-partition. We deal with three problems to find an $(l,u)$-partition of a given graph; the minimum partition problem is to find an $(l,u)$-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p-partition problem is to find an $(l,u)$-partition with a fixed number p of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time $\mathrm{O}(u^{4}n)$ and the p-partition problem can be solved in time $\mathrm{O}(p^2{u}^{4}n)$ for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.