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Lower Bounds for the Happy Coloring Problems

  • Ivan Bliznets
  • Danil SagunovEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11653)

Abstract

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal ’17, Aravind et al. ’16, Choudhari and Reddy ’18, Misra and Reddy ’17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy ’17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an \({\mathcal {O}}^*(\ell ^k)\) randomized algorithm for MHV and an \({\mathcal {O}}^*(2^k)\) algorithm for MHE, where \(\ell \) is the number of colors used and k is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

References

  1. 1.
    Agrawal, A.: On the parameterized complexity of happy vertex coloring. In: Brankovic, L., Ryan, J., Smyth, W.F. (eds.) IWOCA 2017. LNCS, vol. 10765, pp. 103–115. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78825-8_9CrossRefGoogle Scholar
  2. 2.
    Aravind, N.R., Kalyanasundaram, S., Kare, A.S.: Linear time algorithms for happy vertex coloring problems for trees. In: Mäkinen, V., Puglisi, S.J., Salmela, L. (eds.) IWOCA 2016. LNCS, vol. 9843, pp. 281–292. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44543-4_22CrossRefzbMATHGoogle Scholar
  3. 3.
    Aravind, N., Kalyanasundaram, S., Kare, A.S., Lauri, J.: Algorithms and hardness results for happy coloring problems. arXiv preprint arXiv:1705.08282 (2017)
  4. 4.
    Choudhari, J., Reddy, I.V.: On structural parameterizations of happy coloring, empire coloring and boxicity. In: Rahman, M.S., Sung, W.-K., Uehara, R. (eds.) WALCOM 2018. LNCS, vol. 10755, pp. 228–239. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-75172-6_20CrossRefzbMATHGoogle Scholar
  5. 5.
    Cygan, M., et al.: On problems as hard as CNF-SAT. ACM Trans. Algorithms 12(3), 1–24 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cygan, M., et al.: Lower bounds for kernelization. Parameterized Algorithms, pp. 523–555. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3_15CrossRefGoogle Scholar
  7. 7.
    Cygan, M., et al.: Parameterized Algorithms, vol. 3. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21275-3CrossRefzbMATHGoogle Scholar
  8. 8.
    Dell, H., Husfeldt, T., Marx, D., Taslaman, N., Wahlén, M.: Exponential time complexity of the permanent and the Tutte polynomial. ACM Trans. Algorithms 10(4), 1–32 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dell, H., Marx, D.: Kernelization of packing problems. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics (2012)Google Scholar
  10. 10.
    Dell, H., Melkebeek, D.V.: Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses. J. ACM 61(4), 1–27 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Diestel, R.: Graph Theory. Springer, Heidelberg (2018)zbMATHGoogle Scholar
  12. 12.
    Dom, M., Lokshtanov, D., Saurabh, S.: Kernelization lower bounds through colors and IDs. ACM Trans. Algorithms 11(2), 1–20 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gao, H., Gao, W.: Kernelization for maximum happy vertices problem. In: Bender, M.A., Farach-Colton, M., Mosteiro, M.A. (eds.) LATIN 2018. LNCS, vol. 10807, pp. 504–514. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-77404-6_37CrossRefGoogle Scholar
  14. 14.
    Garey, M., Johnson, D., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hermelin, D., Wu, X.: Weak compositions and their applications to polynomial lower bounds for kernelization. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics (2012)Google Scholar
  16. 16.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations, pp. 85–103. Springer, Boston (1972).  https://doi.org/10.1007/978-1-4684-2001-2_9CrossRefGoogle Scholar
  17. 17.
    Lewis, R., Thiruvady, D., Morgan, K.: Finding happiness: an analysis of the maximum happy vertices problem. Comput. Oper. Res. 103, 265–276 (2019)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Misra, N., Reddy, I.V.: The parameterized complexity of happy colorings. In: Brankovic, L., Ryan, J., Smyth, W.F. (eds.) IWOCA 2017. LNCS, vol. 10765, pp. 142–153. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78825-8_12CrossRefGoogle Scholar
  19. 19.
    Xu, Y., Goebel, R., Lin, G.: Submodular and supermodular multi-labeling, and vertex happiness. CoRR (2016)Google Scholar
  20. 20.
    Zhang, P., Jiang, T., Li, A.: Improved approximation algorithms for the maximum happy vertices and edges problems. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 159–170. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-21398-9_13CrossRefGoogle Scholar
  21. 21.
    Zhang, P., Li, A.: Algorithmic aspects of homophyly of networks. Theoret. Comput. Sci. 593, 117–131 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang, P., Xu, Y., Jiang, T., Li, A., Lin, G., Miyano, E.: Improved approximation algorithms for the maximum happy vertices and edges problems. Algorithmica 80(5), 1412–1438 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.St. Petersburg Department of Steklov Institute of Mathematics of the Russian Academy of SciencesSaint PetersburgRussia
  2. 2.National Research University Higher School of EconomicsSaint PetersburgRussia

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