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Approximation Algorithms for Maximally Balanced Connected Graph Partition

Abstract

Given a connected graph \(G = (V, E)\), we seek to partition the vertex set V into k non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these k parts is minimized. We refer this problem to as min-max balanced connected graph partition into k parts and denote it as k -BGP. The vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm. The vertex-weighted 2-BGP and 3-BGP admit a 5/4-approximation and a 3/2-approximation, respectively. When \(k \ge 4\), no approximability result exists for k -BGP, i.e., the vertex unweighted variant, except a trivial k-approximation. In this paper, we present another 3/2-approximation for the 3-BGP and then extend it to become a k/2-approximation for k -BGP, for any fixed \(k \ge 3\). Furthermore, for 4-BGP, we propose an improved 24/13-approximation. To these purposes, we have designed several local improvement operations, which could find more applications in related graph partition problems.

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Notes

  1. 1.

    An extended abstract appears in the Proceedings of COCOA 2019. LNCS 11949, pages 130–141.

  2. 2.

    Basically, we reserve the word “connected” for a graph and the word “adjacent” for two objects with at least one edge between them.

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Acknowledgements

We are very grateful to the anonymous reviewers for their many helpful comments and suggestions to improve the presentation. YC and AZ are supported by the NSFC Grants 11971139, 11771114, and the Zhejiang Provincial NSF Grant LY21A010014; they are also supported by the CSC Grants 201508330054 and 201908330090, respectively. ZZC is supported by in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan, under Grant No. 18K11183. GL is supported by the NSERC Canada.

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Correspondence to Guohui Lin.

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Chen, Y., Chen, ZZ., Lin, G. et al. Approximation Algorithms for Maximally Balanced Connected Graph Partition. Algorithmica 83, 3715–3740 (2021). https://doi.org/10.1007/s00453-021-00870-3

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Keywords

  • Graph partition
  • Induced subgraph
  • Connected component
  • Local improvement
  • Approximation algorithm