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Algorithmica

, Volume 80, Issue 5, pp 1412–1438 | Cite as

Improved Approximation Algorithms for the Maximum Happy Vertices and Edges Problems

  • Peng Zhang
  • Yao Xu
  • Tao Jiang
  • Angsheng Li
  • Guohui Lin
  • Eiji Miyano
Article
Part of the following topical collections:
  1. Special Issue on Computing and Combinatorics

Abstract

The Maximum Happy Vertices (MHV) problem and the Maximum Happy Edges (MHE) problem are two fundamental problems arising in the study of the homophyly phenomenon in large scale networks. Both of these two problems are NP-hard. Interestingly, the MHE problem is a natural generalization of Multiway Uncut, the complement of the classic Multiway Cut problem. In this paper, we present new approximation algorithms for MHV and MHE based on randomized LP-rounding technique and non-uniform approach. Specifically, we show that MHV can be approximated within \(\frac{1}{\varDelta +1/g(\varDelta )}\), where \(\varDelta \) is the maximum vertex degree and \(g(\varDelta ) = (\sqrt{\varDelta } + \sqrt{\varDelta +1})^2 \varDelta \), and MHE can be approximated within \(\frac{1}{2} + \frac{\sqrt{2}}{4}f(k) \ge 0.8535\), where \(f(k) \ge 1\) is a function of the color number k. These results improve over the previous approximation ratios for MHV, MHE as well as Multiway Uncut in the literature.

Keywords

Maximum Happy Vertices Maximum Happy Edges Approximation algorithm Randomized rounding Network homophyly 

Mathematics Subject Classification

68W25 90C27 

Notes

Acknowledgements

We thank the anonymous reviewers for their suggestions which help to improve the presentation of the paper.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Peng Zhang
    • 1
  • Yao Xu
    • 2
  • Tao Jiang
    • 3
    • 4
  • Angsheng Li
    • 5
  • Guohui Lin
    • 2
  • Eiji Miyano
    • 6
  1. 1.School of Computer Science and TechnologyShandong UniversityJinanChina
  2. 2.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
  3. 3.Department of Computer Science and EngineeringUniversity of CaliforniaRiversideUSA
  4. 4.MOE Key Lab of Bioinformatics and Bioinformatics Division, TNLIST/Department of Computer Science and TechnologyTsinghua UniversityBeijingChina
  5. 5.State Key Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingChina
  6. 6.Department of Systems Design and InformaticsKyushu Institute of TechnologyIizukaJapan

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