New and Improved Bounds for the Minimum Set Cover Problem

  • Rishi Saket
  • Maxim Sviridenko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

Abstract

We study the relationship between the approximation factor for the Set-Cover problem and the parameters Δ : the maximum cardinality of any subset, and k : the maximum number of subsets containing any element of the ground set. We show an LP rounding based approximation of \((k-1)(1-e^{-\frac{\ln \Delta}{k-1}}) +1\), which is substantially better than the classical algorithms in the range k ≈ ln Δ, and also improves on related previous works [19,22]. For the interesting case when k = θ(logΔ) we also exhibit an integrality gap which essentially matches our approximation algorithm. We also prove a hardness of approximation factor of \(\Omega\left(\frac{\log \Delta}{(\log\log \Delta)^2}\right)\) when k = θ(logΔ). This is the first study of the hardness factor specifically for this range of k and Δ, and improves on the only other such result implicitly proved in [18].

Keywords

Vertex Cover Linear Programming Relaxation Performance Guarantee Satisfying Assignment Hardness Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Moshkovitz, D., Safra, M.: Algorithmic construction of sets for k-restrictions. The ACM Transactions on Algorithms 2(2), 153–177 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Austrin, P., Mossel, E.: Approximation Resistant Predicates from Pairwise Independence. Computational Complexity 18(2), 249–271 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bansal, N., Khot, S.: Inapproximability of Hypergraph Vertex Cover and Applications to Scheduling Problems. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 250–261. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Bar-Yehuda, R., Even, S.: A linear-time approximation algorithm for the weighted vertex cover problem. J. Algorithms 2(2), 198–203 (1981)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Chvatal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34(5), 1129–1146 (2005)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Guruswami, V., Raghavendra, P.: Constraint Satisfaction over a Non-Boolean Domain: Approximation Algorithms and Unique-Games Hardness. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 77–90. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Halperin, E.: Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs. SIAM J. Comput. 31(5), 1608–1623 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Hochbaum, D.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comput. 11(3), 555–556 (1982)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Holmerin, J.: Improved Inapproximability Results for Vertex Cover on k-Uniform Hypergraphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 1005–1016. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Johnson, D.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)MATHCrossRefGoogle Scholar
  13. 13.
    Khot, S.: Improved inaproximability results for maxclique, chromatic number and approximate graph coloring. In: Proc. FOCS, pp. 600–609 (2001)Google Scholar
  14. 14.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proc. STOC, pp. 767–775 (2002)Google Scholar
  15. 15.
    Khot, S.: Online lecture notes for Probabilistically Checkable Proofs and Hardness of Approximation, Lecture 3 (scribed by Deeparnab Chakrabarty), www.cs.nyu.edu/~khot/pcp-lecnotes/lec3.ps
  16. 16.
    Khot, S., Ponnuswami, A.K.: Better Inapproximability Results for MaxClique, Chromatic Number and Min-3Lin-Deletion. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 226–237. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. J. Comput. System Sci. 74(3), 335–349 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Khot, S., Saket, R.: Hardness of Minimizing and Learning DNF Expressions. In: Proc. FOCS, pp. 231–240 (2008)Google Scholar
  19. 19.
    Krivelevich, M.: Approximate set covering in uniform hypergraphs. J. Algorithms 25(1), 118–143 (1997)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Lovasz, L.: On the ratio of the optimal integral and fractional covers. Disc. Math. 13, 383–390 (1975)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. ACM 31(5), 960–981 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Okun, M.: On the approximation of the vertex cover problem in hypergraphs. Discrete Optimization 2(1), 101–111 (2005)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Raz, R., Safra, M.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. STOC, pp. 475–484 (2007)Google Scholar
  24. 24.
    Samorodnitsky, A., Trevisan, L.: A PCP characterization of NP with optimal amortized query complexity. In: Proc. STOC, pp. 191–199 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rishi Saket
    • 1
  • Maxim Sviridenko
    • 2
  1. 1.IBM T.J. Watson Research CenterUSA
  2. 2.Department of Computer ScienceUniversity of WarwickUK

Personalised recommendations