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

Takehiro ITO  Kazuya GOTO  Xiao ZHOU  Takao NISHIZEKI 

Publication
IEICE TRANSACTIONS on Information and Systems  Vol.E90-D  No.2  pp.449-456
Publication Date: 2007/02/01
Online ISSN: 1745-1361
Print ISSN: 0916-8532
Type of Manuscript: Special Section PAPER (Special Section on Foundations of Computer Science)
Category: Graph Algorithms
Keyword: 
algorithm,  choice partition,  lower bound,  maximum partition problem,  minimum partition problem,  multi-weighted graph,  partial k-tree,  series-parallel graph,  uniform partition,  upper bound, 

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Summary: 
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.