, 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
Part of the following topical collections:
  1. Special Issue on Computing and Combinatorics


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.


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

Mathematics Subject Classification

68W25 90C27 



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


  1. 1.
    Anstreicher, K.M.: Linear programming in \(O(\frac{n^3}{\ln n} L)\) operations. SIAM J. Optim. 9(4), 803–812 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Breiman, L., Friedman, J., Olshen, R., Stone, C.: Classification and Regression Trees. Wadsworth and Brooks, Monterey (1984)zbMATHGoogle Scholar
  3. 3.
    Buchbinder, N., Naor, J., Schwartz, R.: Simplex partitioning via exponential clocks and the multiway cut problem. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), pp. 535–544. Palo Alto, CA, USA (2013)Google Scholar
  4. 4.
    Calinescu, G., Karloff, H., Rabani, Y.: An improved approximation algorithm for multiway cut. J. Comput. Syst. Sci. 60(3), 564–574 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23, 864–894 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  7. 7.
    Karger, D., Klein, P., Stein, C., Thorup, M., Young, N.: Rounding algorithms for a geometric embedding of minimum multiway cut. Math. Oper. Res. 29(3), 436–461 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kleinberg, J., Tardos, É.: Approximation algorithms for classification problems with pairwise relationships: metric labeling and Markov random fields. J. ACM 49(5), 616–639 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Langberg, M., Rabani, Y., Swamy, C.: Approximation algorithms for graph homomorphism problems. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) Proceedings of the 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pp. 176–187. Barcelona, Spain (2006)Google Scholar
  10. 10.
    Li, A., Li, J., Pan, Y.: Homophyly/kinship hypothesis: natural communities, and predicting in networks. Phys. A 420, 148–163 (2015)CrossRefGoogle Scholar
  11. 11.
    Sharma, A., Vondrák, J.: Multiway cut, pairwise realizable distributions, and descending thresholds. In: Shmoys, D. (ed.) Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC), pp. 724–733 (2014)Google Scholar
  12. 12.
    Vazirani, V.: Approximation Algorithms, 2nd edn. Springer, Berlin (2003)CrossRefGoogle Scholar
  13. 13.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Zhang, P., Jiang, T., Li, A.: Improved approximation algorithms for the maximum happy vertices and edges problems. In: Xu, D., Du, D., Du, D.-Z. (eds.) Proceedings of the 21th International Computing and Combinatorics Conference (COCOON), Volume 9198 of LNCS, pp. 159–170 (2015)Google Scholar
  15. 15.
    Zhang, P., Li, A.: Algorithmic aspects of homophyly of networks. Theor. Comput. Sci. 593, 117–131 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations