Approximation Algorithms for Partial Covering Problems

Extended Abstract
  • Rajiv Gandhi
  • Samir Khuller
  • Aravind Srinivasan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2076)

Abstract

We study the generalization of covering problems to partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-set cover, if each element occurs in at most f sets, then we derive a primal-dual f-approximation algorithm (thus implying a 2-approximation for k-vertex cover) in polynomial time. In addition to its simplicity, this algorithm has the advantage of being parallelizable. For instances where each set has cardinality at most three, we obtain an approximation of 4/3. We also present better-than-2-approximation algorithms for k-vertex cover on bounded degree graphs, and for vertex cover on expanders of bounded average degree. We obtain a polynomial-time approximation scheme for k-vertex cover on planar graphs, and for covering points in R d by disks.

Keywords and Phrases

Approximation algorithms partial covering set cover vertex cover primal-dual methods randomized rounding 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N. Alon, R. Boppana and J. H. Spencer. An asymptotic isoperimetric inequality. Geometric and Functional Analysis, 8:411–436, 1998.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    B. Baker. Approximation Algorithms for NP-Complete Problems on Planar Graphs. JACM, Vol 41 (1), (1994), pp. 153–190.MATHCrossRefGoogle Scholar
  3. 3.
    R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. J. of Algorithms 2:198–203, 1981.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. Bar-Yehuda and S. Even. A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25:27–45, 1985.MathSciNetGoogle Scholar
  5. 5.
    R. Bar-Yehuda. Using homogeneous weights for approximating the partial cover problem. In Proc. Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, 71–75, 1999.Google Scholar
  6. 6.
    N. Bshouty, and L. Burroughs. Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. The Proceedings of the Fifteenth Annual Symposium on the Theoretical Aspects of Computer Science 298–308, 1998.Google Scholar
  7. 7.
    M. Charikar, S. Khuller, D. Mount, and G. Narasimhan. Algorithms for Facility Location Problems with Outliers. In Proc. Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, 642–651, 2001.Google Scholar
  8. 8.
    V. Chvátal. A greedy heuristic for the set-covering problem. Math. of Oper. Res. Vol. 4, 3, 233–235, 1979.MATHCrossRefGoogle Scholar
  9. 9.
    K. L. Clarkson. A modification of the greedy algorithm for the vertex cover. Information Processing Letters 16:23–25, 1983.CrossRefMathSciNetGoogle Scholar
  10. 10.
    T. H. Cormen, C. E. Leiserson and R. L. Rivest, “Introduction to Algorithms”, MIT Press, 1989.Google Scholar
  11. 11.
    R. Duh and M. Fürer. Approximating k-set cover by semi-local optimization. In Proc. 29th STOC, May 1997, pages 256–264.Google Scholar
  12. 12.
    R. Gandhi, S. Khuller and A. Srinivasan. Approximation algorithms for partial covering problems. Technical Report CS-TR-# 4234 (April 2001). Also available at: http://www.cs.umd.edu/users/samir/grant/icalp01.ps
  13. 13.
    O. Goldschmidt, D. Hochbaum, and G. Yu. A modified greedy heuristic for the set covering problem with improved worst case bound. Information Processing Letters 48(1993), 305–310.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. X. Goemans and J. Kleinberg. The Lovász theta function and a semidefinite programming relaxation of vertex cover. SIAM Journal on Discrete Mathematics, 11:196–204, 1998.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24:296–317, 1995.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Halldórsson. Approximating k-set cover and complementary graph coloring. In Proc. Fifth Conference on Integer Programming and Combinatorial Optimization, June 1996, LNCS 1084, pages 118–131.Google Scholar
  17. 17.
    E. Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In Proc. Eleventh ACM-SIAM Symposium on Discrete Algorithms, January 2000, pages 329–337.Google Scholar
  18. 18.
    D. S. Hochbaum. Approximation algorithms for the set covering and vertex cover problems. W.P.#64-79-80, GSIA, Carnegie-Mellon University, April 1980. Also: SIAM J. Comput. 11(3) 1982.Google Scholar
  19. 19.
    D. S. Hochbaum. Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6:243–254, 1983.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    D. S. Hochbaum (editor). Approximation Algorithms for NP-hard problems. PWS Publishing Company, 1996.Google Scholar
  21. 21.
    D. S. Hochbaum. The t-vertex cover problem: Extending the half integrality framework with budget constraints. In Proc. First International Workshop on Approximation Algorithms for Combinatorial Optimization Problems 111–122, 1998.Google Scholar
  22. 22.
    D. S. Hochbaum and W. Maass. Approximation schemes for covering and packing problems in image processing and VLSI. Journal of ACM, 32(1):130–136, 1985.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    D. S. Johnson. Approximation algorithms for combinatorial problems. J. Comput. System Sci., 9:256–278, 1974.MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    M. Kearns. The computational complexity of machine learning. M.I.T. Press, 1990.Google Scholar
  25. 25.
    S. Khuller, U. Vishkin, and N. Young. A Primal Dual Parallel Approximation Technique Applied to Weighted Set and Vertex Cover. Journal of Algorithms, 17(2):280–289, 1994.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Math. 13:383–390, 1975.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    G. L. Nemhauser and L. E. Trotter, Jr. Vertex packings: Structural properties and algorithms. Mathematical Programming 8:232–248, 1975.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    E. Petrank. The hardness of approximation: Gap location. Computational Complexity 4:133–157, 1994.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    P. Slavík. Improved performance of the greedy algorithm for partial cover. Information Processing Letters 64:251–254, 1997.CrossRefMathSciNetGoogle Scholar
  30. 30.
    A. Srinivasan. New Approaches to Covering and Packing Problems. In Proc. Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, 567–576, 2001.Google Scholar

Copyright information

© Sprunger-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Rajiv Gandhi
    • 1
  • Samir Khuller
    • 2
  • Aravind Srinivasan
    • 3
  1. 1.Department of Computer ScienceUniversity of MarylandMD
  2. 2.Department of Computer Science and Institute for Advanced Computer StudiesUniversity of MarylandMD
  3. 3.Bell LabsLucent TechnologiesMurray Hill

Personalised recommendations