Advertisement

Journal of Statistical Physics

, Volume 159, Issue 5, pp 1154–1174 | Cite as

Statistical Mechanics of the Minimum Dominating Set Problem

  • Jin-Hua Zhao
  • Yusupjan Habibulla
  • Hai-Jun ZhouEmail author
Article

Abstract

The minimum dominating set (MDS) problem has wide applications in network science and related fields. It aims at constructing a node set of smallest size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this optimization problem is generally very difficult, we show it can be exactly solved by a generalized leaf-removal (GLR) process if the network contains no core. We present a percolation theory to describe the GLR process on random networks, and solve a spin glass model by mean field method to estimate the MDS size. We also implement a message-passing algorithm and a local heuristic algorithm that combines GLR with greedy node-removal to obtain near-optimal solutions for single random networks. Our algorithms also perform well on real-world network instances.

Keywords

Dominating set Spin glass Core percolation Leaf removal  Network coarse-graining Belief propagation 

Notes

Acknowledgments

Part of this work was done when H.-J. Zhou was participating in the “Collective Dynamics in Information Systems 2014” Program of the Kavli Institute for Theoretical Physics China (KITPC). H.-J. Zhou thanks Chuang Wang for a helpful discussion, and Alfredo Braunstein, Yang-Yu Liu, Federico Ricci-Tersenghi, and Yi-Fan Sun for helpful comments on the manuscript; J.-H. Zhao and H.-J. Zhou thank Prof. Zhong-Can Ou-Yang for support. Research partially supported by the National Basic Research Program of China (grant number 2013CB932804) and by the National Natural Science Foundation of China (grand numbers 11121403 and 11225526).

References

  1. 1.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  2. 2.
    Yang, Y., Wang, J., Motter, A.E.: Network observability transitions. Phys. Rev. Lett. 109, 258701 (2012)CrossRefADSGoogle Scholar
  3. 3.
    Biroli, G., Mézard, M.: Lattice glass models. Phys. Rev. Lett. 88, 025501 (2002)CrossRefADSGoogle Scholar
  4. 4.
    Hartmann, A.K., Weigt, M.: Statistical mechanics of the vertex-cover problem. J. Phys. A 36, 11069–11093 (2003)CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. 5.
    Zhao, J.-H., Zhou, H.-J.: Statistical physics of hard combinatorial optimization: vertex cover problem. Chin. Phys. B 23, 078901 (2014)CrossRefADSGoogle Scholar
  6. 6.
    Echenique, P., Gómez-Gardeñes, J., Moreno, Y., Vázquez, A.: Distance-\(d\) covering problems in scale-free networks with degree correlations. Phys. Rev. E 71, 035102(R) (2005)CrossRefADSGoogle Scholar
  7. 7.
    Takaguchi, T., Hasegawa, T., Yoshida, Y.: Suppressing epidemics on networks by exploiting observer nodes. Phys. Rev. E 90, 012807 (2014)CrossRefADSGoogle Scholar
  8. 8.
    Liu, Y.-Y., Slotine, J.-J., Barabási, A.-L.: Controllability of complex networks. Nature 473, 167–173 (2011)CrossRefADSGoogle Scholar
  9. 9.
    Wuchty, S.: Controllability in protein interaction networks. Proc. Natl. Acad. Sci. USA 111, 7156–7160 (2014)CrossRefADSGoogle Scholar
  10. 10.
    Wu, J., Li, H.: A dominating-set-based routing scheme in ad hoc wireless networks. Telecomm. Syst. 18, 13–36 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Bramoulle, Y., Kranton, R.: Public goods in networks. J. Econom. Theor. 135, 478–494 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Shen, C., Li, T.: Multi-document summarization via the minimum dominating set. In: Proceedings of the 23rd International Conference on Computational Linguistics (Coling 2010, Beijing), pp. 984–992 (Association for Computational Linguistics, 2010)Google Scholar
  13. 13.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 41, 960–981 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability pcp characterization of np. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 475–484 (ACM, New York, 1997)Google Scholar
  15. 15.
    Hedar, A.-R., Ismail, R.: Simulated annealing with stochastic local search for minimum dominating set problem. Int. J. Mach. Learn. Cybernet. 3, 97–109 (2012)CrossRefGoogle Scholar
  16. 16.
    Molnár Jr, F., Sreenivasan, S., Szymanski, B.K., Korniss, K.: Minimum dominating sets in scale-free network ensembles. Sci. Rep. 3, 1736 (2013)CrossRefADSGoogle Scholar
  17. 17.
    Bauer, M., Golinelli, O.: Core percolation in random graphs: a critical phenomena analysis. Eur. Phys. J. B 24, 339–352 (2001)CrossRefADSGoogle Scholar
  18. 18.
    Lucibello, C., Ricci-Tersenghi, F.: The statistical mechanics of random set packing and a generalization of the Karp–Sipser algorithm. Int. J. Stat. Mech. 2014, 136829 (2014)CrossRefGoogle Scholar
  19. 19.
    Takabe, S., Hukushima, K.: Minimum vertex cover problems on random hypergraphs: replica symmetric solution and a leaf removal algorithm. Phys. Rev. E 89, 062139 (2014)CrossRefADSGoogle Scholar
  20. 20.
    He, D.-R., Liu, Z.-H., Wang, B.-H.: Complex Systems and Complex Networks. Higher Education Press, Beijing (2009)zbMATHGoogle Scholar
  21. 21.
    Albert, R., Barabási, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)CrossRefADSzbMATHGoogle Scholar
  22. 22.
    Goh, K.-I., Kahng, B., Kim, D.: Universal behavior of load distribution in scale-free networks. Phys. Rev. Lett. 87, 278701 (2001)CrossRefADSGoogle Scholar
  23. 23.
    Zhao, J.-H., Zhou, H.-J., Liu, Y.-Y.: Inducing effect on the percolation transition in complex networks. Nat. Commun. 4, 2412 (2013)ADSGoogle Scholar
  24. 24.
    Zhou, H.J.: Long-range frustration in a spin-glass model of the vertex-cover problem. Phys. Rev. Lett. 94, 217203 (2005)CrossRefADSGoogle Scholar
  25. 25.
    Zhou, H.-J.: Erratum: long-range frustration in a spin-glass model of the vertex-cover problem [phys. rev. lett. 94, 217203 (2005)]. Phys. Rev. Lett. 109, 199901 (2012)CrossRefADSGoogle Scholar
  26. 26.
    Catanzaro, M., Pastor-Satorras, R.: Analytic solution of a static scale-free network model. Eur. Phys. J. B 44, 241–248 (2005)CrossRefADSGoogle Scholar
  27. 27.
    Zhou, H.J., Lipowsky, R.: Dynamic pattern evolution on scale-free networks. Proc. Natl. Acad. Sci. USA 102, 10052–10057 (2005)CrossRefADSGoogle Scholar
  28. 28.
    Šubelj, L., Bajec, M.: Robust network community detection using balanced propagation. Eur. Phys. J. B 81, 353–362 (2011)CrossRefADSGoogle Scholar
  29. 29.
    Leskovec, J., Lang, K.J., Dasgupta, A., Mahoney, M.W.: Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Math. 6, 29–123 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ netowrks. Nature 393, 440–442 (1998)CrossRefADSGoogle Scholar
  31. 31.
    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graphs over time: densification laws, shrinking diameters and possible explanations. In: Proceedings of the Eleventh ACM SIGKDD International Conference on Knowledge Discovery in Data Mining, pp. 177–187 (ACM, New York, 2005)Google Scholar
  32. 32.
    Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: densification and shrinking diameters. ACM Trans. Knowl. Discov. Data 1, 2 (2007)CrossRefGoogle Scholar
  33. 33.
    Ripeanu, M., Foster, I., Iamnitchi, A.: Mapping the gnutella network: properties of large-scale peer-to-peer systems and implications for system design. IEEE Internet Comput. 6, 50–57 (2002)Google Scholar
  34. 34.
    Cho, E., Myers, S. A., Leskovec, J.: Friendship and mobility: user movement in localation-based social networks. In: ACM SIGKDD International Conference o Knowledge Discovery and Data Mining, pp. 1082–1090 (San Diego, CA, USA, 2011)Google Scholar
  35. 35.
    Bu, D., et al.: Topological structure analysis of the protein–protein interaction network in budding yeast. Nucleic Acids Res. 31, 2443–2450 (2003)CrossRefGoogle Scholar
  36. 36.
    Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford Univ. Press, New York (2009)CrossRefzbMATHGoogle Scholar
  37. 37.
    Xiao, J.-Q., Zhou, H.J.: Partition function loop series for a general graphical model: free-energy corrections and message-passing equations. J. Phys. A 44, 425001 (2011)CrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Zhou, H.J., Wang, C.: Region graph partition function expansion and approximate free energy landscapes: theory and some numerical results. J. Stat. Phys. 148, 513–547 (2012)CrossRefADSzbMATHMathSciNetGoogle Scholar
  39. 39.
    Mézard, M., Parisi, G., Zecchina, R.: Analytic and algorithmic solution of random satisfiability problems. Science 297, 812–815 (2002)CrossRefADSGoogle Scholar
  40. 40.
    Krzakala, F., Montanari, A., Ricci-Tersenghi, F., Semerjian, G., Zdeborova, L.: Gibbs states and the set of solutions of random constraint satisfaction problems. Proc. Natl. Acad. Sci. USA 104, 10318–10323 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  41. 41.
    Altarelli, F., Braunstein, A., Dall’Asta, L., Zecchina, R.: Large deviations of cascade processes on graphs. Phys. Rev. E 87, 062115 (2013)CrossRefADSGoogle Scholar
  42. 42.
    Altarelli, F., Braunstein, A., Dall’Asta, L., Zecchina, R.: Optimizing spread dynamics on graphs by message passing. J. Stat. Mech. (2013). doi: 10.1088/1742-5468/2013/09/P09011
  43. 43.
    Guggiola, A., Semerjian, G.: Minimal contagious sets in random regular graphs. J. Stat. Phys. 158, 300–358 (2015)CrossRefADSMathSciNetGoogle Scholar
  44. 44.
    Hasegawa, T., Takaguchi, T., Masuda, N.: Observability transitions in correlated networks. Phys. Rev. E 88, 042809 (2013)CrossRefADSGoogle Scholar
  45. 45.
    Mézard, M., Parisi, G.: The bethe lattice spin glass revisited. Eur. Phys. J. B 20, 217–233 (2001)CrossRefADSGoogle Scholar
  46. 46.
    Mézard, M., Montanari, A.: Reconstruction on trees and spin glass transition. J. Stat. Phys. 124, 1317–1350 (2006)CrossRefADSzbMATHMathSciNetGoogle Scholar
  47. 47.
    Habibulla, Y., Zhao, J.-H., Zhou, H.-J.: The directed dominating set problem: generalized leaf removal and belief propagation. (2015, in preparation)Google Scholar
  48. 48.
    Du, D.-Z., Wan, P.-J.: Connected Dominating Set: Theory and Applications. Springer, New York (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jin-Hua Zhao
    • 1
  • Yusupjan Habibulla
    • 1
  • Hai-Jun Zhou
    • 1
    Email author
  1. 1.State Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsChinese Academy of SciencesBeijingChina

Personalised recommendations