Abstract
Given a simple 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 general 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; but no approximability result exists for k -BGP when \(k \ge 4\), except a trivial k-approximation. In this paper, we present another 3/2-approximation for our cardinality 3-BGP and then extend it to become a k/2-approximation for k -BGP, for any constant \(k \ge 3\). Furthermore, for 4-BGP, we propose an improved 24/13-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems.
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Notes
- 1.
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
CY and AZ are supported by the NSFC Grants 11971139, 11771114 and 11571252; and 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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Chen, Y., Chen, ZZ., Lin, G., Xu, Y., Zhang, A. (2019). Approximation Algorithms for Maximally Balanced Connected Graph Partition. In: Li, Y., Cardei, M., Huang, Y. (eds) Combinatorial Optimization and Applications. COCOA 2019. Lecture Notes in Computer Science(), vol 11949. Springer, Cham. https://doi.org/10.1007/978-3-030-36412-0_11
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