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Unbalanced Graph Partitioning

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Abstract

We investigate the unbalanced cut problems. A cut (A,B) is called unbalanced if the size of its smaller side is at most k (called k-size) or exactly k (called Ek-size), where k is an input parameter. We consider two closely related unbalanced cut problems, in which the quality of a cut is measured with respect to the sparsity and the conductance, respectively.

We show that even if the input graph is restricted to be a tree, the Ek-Sparsest Cut problem (to find an Ek-size cut with the minimum sparsity) is still NP-hard. We give a bicriteria approximation algorithm for the k-Sparsest Cut problem (to find a k-size cut with the minimum sparsity), which outputs a cut whose sparsity is at most O(logn) times the optimum and whose smaller side has size at most O(logn)k. As a consequence, this leads to a (O(logn),O(logn))-bicriteria approximation algorithm for the Min k-Conductance problem (to find a k-size cut with the minimum conductance).

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Acknowledgements

We are very grateful to the anonymous referee for the valuable comments which make the presentation of the paper brief and clear. We thank Wei Chen, Pinyan Lu and Yajun Wang (Theory Group, Microsoft Research Asia) for helpful discussions on the paper.

Part of the second author’s work was done when the author visited Microsoft Research Asia. Angsheng Li is supported by the hundred talent program of the Chinese Academy of Sciences, and the grand challenge program, Network Algorithms and Digital Information, Institute of Software, Chinese Academy of Sciences. Peng Zhang is supported by National Natural Science Foundation of China under grant No. 60970003, the StarTrack Program of Microsoft Research Asia, and China Postdoctoral Science Foundation 20080441144, 200902562.

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Authors and Affiliations

  1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Angsheng Li

  2. School of Computer Science and Technology, Shandong University, Jinan, 250101, China

    Peng Zhang

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  1. Angsheng Li
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  2. Peng Zhang
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Correspondence to Peng Zhang.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 21st International Symposium on Algorithms and Computation (ISAAC), Part I, pages 218–229 [11].

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Li, A., Zhang, P. Unbalanced Graph Partitioning. Theory Comput Syst 53, 454–466 (2013). https://doi.org/10.1007/s00224-012-9436-x

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