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 O(u 4 n) and the p-partition problem can be solved in time 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.

Topology Control in Large Scale WSNs : Routing and Base Station Placement

“…In (Ito et al, 2006), authors considered three problems to find a (l, u)-partition of a given graph. They proposed to partition a graph G into connected components by deleting some edges from G making the total weight of each component equal at least to l and at most to u.…”

Multiple Mobile Sinks Deployment for Energy Efficiency in Large Scale Wireless Sensor Networks

“…They define a ρ-separator as a sub-set of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most ρ times the weight of the graph. In [21], authors considered three problems to find an (l, u)-partition of a given graph. They proposed to partition a graph G into connected components by deleting some edges from G making the total weight of each component equal at least to l and at most to u.…”

Partitioning a Weighted Tree into Subtrees with Weights in a Given Range

“…Since K 2,n−2 is a series-parallel graph, the three partition problems are NP-hard even for seriesparallel graphs [8]. Therefore, it is very unlikely that the three partition problems can be solved in polynomial time even for series-parallel graphs, although the three problems can be solved in pseudo-polynomial time for graphs of bounded tree-width, including series-parallel graphs [8].…”

“…Since K 2,n−2 is a series-parallel graph, the three partition problems are NP-hard even for seriesparallel graphs [8]. Therefore, it is very unlikely that the three partition problems can be solved in polynomial time even for series-parallel graphs, although the three problems can be solved in pseudo-polynomial time for graphs of bounded tree-width, including series-parallel graphs [8]. On the other hand, all the three partition problems can be solved in linear time for paths [10], and a related minimization problem can be solved in polynomial time for special classes of trees such as stars, worms, caterpillars and so on [4].…”