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Approximation Algorithms for Optimization Problems in Random Power-Law Graphs

  • Yilin Shen
  • Xiang Li
  • My T. Thai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

Abstract

Many large-scale real-world networks are well-known to have the power law distribution in their degree sequences: the number of vertices with degree i is proportional to \(i^{-\beta }\) for some constant \(\beta \). It is a common belief that solving optimization problems in power-law graphs is easier. Unfortunately, many problems have been proven NP-hard, along with their inapproximability factors in power-law graphs. Therefore, it is of great importance to develop an algorithm framework such that these optimization problems can be approximated in power-law graphs, with provable theoretical approximation ratios.

In this paper, we propose an algorithmic framework, called Low-Degree Percolation (LDP) Framework, for solving Minimum Dominating Set, Minimum Vertex Cover and Maximum Independent Set problems in power-law graphs. Using this framework, we further show a theoretical framework to derive the approximation ratios for these optimization problems in two well-known random power-law graphs. Our numerical analysis shows that, these optimization problems can be approximated into near 1 factor with high probability, using our proposed LDP algorithms, in power-law graphs with exponential factor \(\beta \ge 1.5\), which belongs to the range of most real-world networks.

Keywords

Power-law graphs Random graphs Approximation algorithms Probabilistic analysis 

Notes

Acknowledgment

This work is partially supported by the NSF CCF-1422116 and DTRA YIP HDTRA-1-09-1-0061 grants.

References

  1. 1.
    Aiello, W., Chung, F., Lu, L.: A random graph model for massive graphs. In: STOC ’00, pp. 171–180. ACM, New York (2000)Google Scholar
  2. 2.
    Aiello, W., Chung, F., Lu, L.: A random graph model for power law graphs. Exp. Math. 10, 53–66 (2000)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Albert, R., Jeong, H., Barabasi, A.L.: The diameter of the world wide web. Nature 401, 130–131 (1999)CrossRefGoogle Scholar
  4. 4.
    Bornholdt, S., Schuster, H.G. (eds.): Handbook of Graphs and Networks: From the Genome to the Internet. Wiley, New York (2003)Google Scholar
  5. 5.
    Chung, F., Lu, L.: Connected components in random graphs with given expected degree sequences. Ann. Comb. 6(2), 125–145 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Eubank, S., Kumar, V.S.A., Marathe, M.V., Srinivasan, A., Wang, N.: Structural and algorithmic aspects of massive social networks. In: SODA ’04, pp. 718–727. Society for Industrial and Applied Mathematics, Philadelphia (2004)Google Scholar
  8. 8.
    Faloutsos, M., Faloutsos, P., Faloutsos, C.: On power-law relationships of the internet topology. In: Proceedings of the Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, SIGCOMM ’99, pp. 251–262. ACM, New York (1999)Google Scholar
  9. 9.
    Ferrante, A., Pandurangan, G., Park, K.: On the hardness of optimization in power-law graphs. Theoret. Comput. Sci. 393(1–3), 220–230 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Gkantsidis, C., Mihail, M., Saberi, A.: Conductance and congestion in power law graphs. SIGMETRICS Perform. Eval. Rev. 31(1), 148–159 (2003)CrossRefGoogle Scholar
  11. 11.
    Halldórsson, M., Radhakrishnan, J.: Greed is good: approximating independent sets in sparse and bounded-degree graphs. In: Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, STOC ’94, pp. 439–448. ACM, New York (1994)Google Scholar
  12. 12.
    Janson, S., Luczak, T., Norros, I.: Large cliques in a power-law random graph(2009)Google Scholar
  13. 13.
    Karakostas, G.: A better approximation ratio for the vertex cover problem. ACM Trans. Algorithms 5(4), 41:1–41:8 (2009)CrossRefMathSciNetGoogle Scholar
  14. 14.
  15. 15.
    Redner, S.: How popular is your paper? an empirical study of the citation distribution. Eur. Phys. J. B - Condens. Matter Complex Syst. 4(2), 131–134 (1998)CrossRefGoogle Scholar
  16. 16.
    Shen, Y., Nguyen, D.T., Xuan, Y., Thai, M.T.: New techniques for approximating optimal substructure problems in power-law graphs. Theoret. Comput. Sci. 447, 107–119 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Vazirani, V.V.: Approximation Algorithms. Springer-Verlag New York Inc., New York (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Samsung Research AmericaSan JoseUSA
  2. 2.CISE DepartmentUniversity of FloridaGainesvilleUSA

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