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Max-min weight balanced connected partition

Author

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  • Lele Wang
  • Zhao Zhang
  • Di Wu
  • Weili Wu
  • Lidan Fan

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. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Lele Wang & Zhao Zhang & Di Wu & Weili Wu & Lidan Fan, 2013. "Max-min weight balanced connected partition," Journal of Global Optimization, Springer, vol. 57(4), pages 1263-1275, December.
    • Handle: RePEc:spr:jglopt:v:57:y:2013:i:4:p:1263-1275
    DOI: 10.1007/s10898-012-0028-8
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    References listed on IDEAS

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    1. Maravalle, Maurizio & Simeone, Bruno & Naldini, Rosella, 1997. "Clustering on trees," Computational Statistics & Data Analysis, Elsevier, vol. 24(2), pages 217-234, April.
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    Cited by:

    1. Guangting Chen & Yong Chen & Zhi-Zhong Chen & Guohui Lin & Tian Liu & An Zhang, 2022. "Approximation algorithms for the maximally balanced connected graph tripartition problem," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1753-1773, October.
    2. Guangting Chen & Yong Chen & Zhi-Zhong Chen & Guohui Lin & Tian Liu & An Zhang, 0. "Approximation algorithms for the maximally balanced connected graph tripartition problem," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-21.

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