Advertisement

Hardness of Set Cover with Intersection 1

  • V. S. Anil Kumar
  • Sunil Arya
  • H. Ramesh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1853)

Abstract

We consider a restricted version of the general Set Covering problem in which each set in the given set system intersects with any other set in at most 1 element. We show that the Set Covering problem with intersection 1 cannot be approximated within a o(logn) factor in random polynomial time unless NP ⊆ ZTIME(n o(log logn)). We also observe that the main challenge in derandomizing this reduction lies in finding a hitting set for large volume combinatorial rectangles satisfying certain intersection properties. These properties are not satisfied by current methods of hitting set construction.

An example of a Set Covering problem with the intersection 1 property is the problem of covering a given set of points in two or higher dimensions using straight lines; any two straight lines intersect in at most one point. The best approximation algorithm currently known for this problem has an approximation factor of θ(logn), and beating this bound seems hard. We observe that this problem is Max-SNP-Hard.

Keywords

Polynomial Time Proof System Approximation Factor Full Version Auxiliary System 
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.
    N. Alon, O. Goldreich, J. Hastad, R. Perralta. Simple Constructions of Almost k-Wise Independent Random Variables. Random Structures and Algorithms, 3, 1992.Google Scholar
  2. 2.
    V.S. Anil Kumar and H. Ramesh. Covering Rectilinear Polygons with Axis-Parallel Rectangles. Proceedings of 31st ACM-SIAM Symposium in Theory of Computing, 1999.Google Scholar
  3. 3.
    S. Arora, C. Lund. Hardness of Approximation. In Approximation Algorithms for NP-Hard Problems, Ed. D. Hochbaum, PWS Publishers, 1995, pp. 399–446.Google Scholar
  4. 4.
    S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy. Proof Verification and Intractability of Approximation Problems. Proceedings of 33rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 13–22.Google Scholar
  5. 5.
    S. Arora, M. Sudan. Improved Low Degree Testing and Applications. Proceedings of the ACM Symposium on Theory of Computing, 1997, pp. 485–495.Google Scholar
  6. 6.
    J. Beck. An Algorithmic Approach to the Lovasz Local Lemma I, Random Structures and Algorithms, 2, 1991, pp. 343–365.MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Bellare, S. Goldwasser, C. Lund, A. Russell. Efficient Probabilistically Checkable Proofs and Applications to Approximation, Proceedings of 25th ACM Symposium on Theory of Computing, 1993, pp. 294–303.Google Scholar
  8. 8.
    H. Brönnimann, M. Goodrich. Almost Optimal Set Covers in Finite VC-Dimension. Discrete Comput. Geom., 14, 1995, pp. 263–279.CrossRefGoogle Scholar
  9. 9.
    Y. Cheng, S.S. Iyengar and R.L. Kashyap. A New Method for Image compression using Irreducible Covers of Maximal Rectangles. IEEE Transactions on Software Engineering, Vol. 14,5, 1988, pp. 651–658.CrossRefGoogle Scholar
  10. 10.
    U. Feige. A threshold of ln n for Approximating Set Cover. Journal of the ACM, 45,4, 1998, pp. 634–652.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    R. Hassin and N. Megiddo. Approximation Algorithms for Hitting Objects with Straight Lines. Discrete Applied Mathematics, 30, 1991, pp. 29–42.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D.S. Johnson. Approximation Algorithms for Combinatorial Problems. Journal of Computing and Systems Sciences, 9, 1974, pp. 256–278.MATHCrossRefGoogle Scholar
  13. 13.
    C. Levcopoulos. Improved Bounds for Covering General Polygons by Rectangles. Proceedings of 6th Foundations of Software Tech. and Theoretical Comp. Sc. LNCS 287, 1987.Google Scholar
  14. 14.
    N. Linial, M. Luby, M. Saks, D. Zuckerman. Hitting Sets for Combinatorial Rectangles. Proceedings of 25 ACM Symposium on Theory of Computing, 1993, pp. 286–293.Google Scholar
  15. 15.
    C. Lund, M. Yannakakis. On the Hardness of Approximating Minimization Problems. Proceedings of 25th ACM Symposium on Theory of Computing, 1993, pp. 286–293.Google Scholar
  16. 16.
    N. Megiddo and A. Tamir, On the complexity of locating linear facilities in the plane, Oper. Res. Let, 1, 1982, pp. 194–197.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    M. Naor, L. Schulman, A. Srinivasan. Splitters and Near-Optimal Derandomization. Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, 1995, pp. 182–191.Google Scholar
  18. 18.
    R. Raz. A Parallel Repetition Theorem. Proceedings of the 27th ACM Symposium on Theory of Computing, 1995, pp. 447–456.Google Scholar
  19. 19.
    R. Raz and S. Safra. A Sub-Constant Error-Probability Low-Degree test and a Sub-Constant Error-Probability PCP Characterization of NP. Proceedings of the ACM Symposium on Theory of Computing, 1997, pp. 475–484.Google Scholar
  20. 20.
    Madhu Sudan. Personal communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • V. S. Anil Kumar
    • 1
  • Sunil Arya
    • 2
  • H. Ramesh
    • 3
  1. 1.MPI für InformatikSaarbrücken
  2. 2.Department of Computer ScienceHong Kong University of Science andTechnologyHong Kong
  3. 3.Department of Computer Science and AutomationIndian Institute of ScienceBangalore

Personalised recommendations