SDP-Based Algorithms for Maximum Independent Set Problems on Hypergraphs

  • Geir Agnarsson
  • Magnús M. Halldórsson
  • Elena Losievskaja
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5555)

Abstract

This paper deals with approximations of maximum independent sets in non-uniform hypergraphs of low degree. We obtain the first performance ratio that is sublinear in terms of the maximum or average degree of the hypergraph. We extend this to the weighted case and give a \(O(\bar{D} \log\log \bar{D}/\log \bar{D})\) bound, where \(\bar{D}\) is the average weighted degree in a hypergraph, matching the best bounds known for the special case of graphs. Our approach is to use an semi-definite technique to sparsify a given hypergraph and then apply combinatorial algorithms to find a large independent set in the resulting sparser instance.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Kahale, N.: Approximating the independence number via the θ-function. Math. Prog. 80(3), 253–264 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bazgan, C., Monnot, J., Paschos, V., Serrière, F.: On the differential approximation of MIN SET COVER. Theor. Comp. Science 332, 497–513 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Caro, Y., Tuza, Z.: Improved lower bounds on k-independence. J. Graph Theory 15, 99–107 (1991)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Feige, U.: Approximating maximum clique by removing subgraphs. SIAM J. Disc. Math. 18(2), 219–225 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Halldórsson, M.M.: Approximations of independent sets in graphs. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 1–13. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Halldórsson, M.M.: Approximations of Weighted Independent Set and hereditary subset problems. J. Graph Algorithms and Applications 4(1), 1–16 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Halldórsson, M.M., Losievskaja, E.: Independent sets in bounded-degree hypergraphs. Disc. Appl. Math. 157, 1773–1786 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Halperin, E.: Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. SIAM J. Computing 31(5), 1608–1623 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. J. ACM 45(2), 246–265 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Krivelevich, M., Nathaniel, R., Sudakov, B.: Approximating coloring and maximum independent sets in 3-uniform hypergraphs. J. Algorithms 41(1), 99–113 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lovász, L.: On Shannon capacity of a graph. IEEE Trans. Inform. Theory 25(1), 1–7 (1979)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mahajan, S., Ramesh, H.: Derandomizing semidefinite programming based approximation algorithms. SIAM J. Computing 28(5), 1641–1663 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rényi, A.: Probability theory. Elsevier, New York (1970)MATHGoogle Scholar
  15. 15.
    Shachnai, H., Srinivasan, A.: Finding large independent sets of hypergraphs in parallel. SIAM J. Disc. Math. 18(3), 488–500 (2005)CrossRefMATHGoogle Scholar
  16. 16.
    Thiele, T.: A lower bound on the independence number of arbitrary hypergraphs. J. Graph Theory 32, 241–249 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Geir Agnarsson
    • 1
  • Magnús M. Halldórsson
    • 2
  • Elena Losievskaja
    • 3
  1. 1.Dept. of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  2. 2.School of Computer ScienceReykjavík UniversityReykjavikIceland
  3. 3.Faculty of Industrial, Mechanical Engineering and Computer ScienceUniversity of IcelandReykjavikIceland

Personalised recommendations