Journal of Global Optimization

, Volume 57, Issue 4, pp 1263–1275 | Cite as

Max-min weight balanced connected partition

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Abstract

For a connected graph \(G=(V,E)\) and a positive integral vertex weight function \(w\), a max-min weight balanced connected \(k\)-partition of \(G\), denoted as \(BCP_k\), is a partition of \(V\) into \(k\) disjoint vertex subsets \((V_1,V_2,\ldots ,V_k)\) such that each \(G[V_i]\) (the subgraph of \(G\) induced by \(V_i\)) is connected, and \(\min _{1\le i\le k}\{w(V_i)\}\) is maximum. Such a problem has a lot of applications in image processing and clustering, and was proved to be NP-hard. In this paper, we study \(BCP_k\) on a special class of graphs: trapezoid graphs whose maximum degree is bounded by a constant. A pseudo-polynomial time algorithm is given, based on which an FPTAS is obtained for \(k=2,3,4\). A step-stone for the analysis of the FPTAS depends on a lower bound for the optimal value of \(BCP_k\) in terms of the total weight of the graph. In providing such a lower bound, a byproduct of this paper is that any 4-connected trapezoid graph on at least seven vertices has a 4-contractible edge, which may have a value in its own right.

Keywords

Balanced connected partition Pseudo-polynomial time algorithm FPTAS 

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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Lele Wang
    • 1
  • Zhao Zhang
    • 1
  • Di Wu
    • 1
  • Weili Wu
    • 2
  • Lidan Fan
    • 2
  1. 1.College of Mathematics and System SciencesXinjiang UniversityUrumqiPeople’s Republic of China
  2. 2.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

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