An Improved Approximation Algorithm for Vertex Cover with Hard Capacities

Extended Abstract
  • Rajiv Gandhi
  • Eran Halperin
  • Samir Khuller
  • Guy Kortsarz
  • Aravind Srinivasan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2719)

Abstract

In this paper we study the capacitated vertex cover problem, a generalization of the well-known vertex cover problem. Given a graph G = (V, E), the goal is to cover all the edges by picking a minimum cover using the vertices. When we pick a vertex, we can cover up to a pre-specified number of edges incident on this vertex (its capacity). The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. Previously, 2-approximation algorithms were developed with the assumption that multiple copies of a vertex may be chosen in the cover. If we are allowed to pick at most a given number of copies of each vertex, then the problem is significantly harder to solve. Chuzhoy and Naor (Proc. IEEE Symposium on Foundations of Computer Science, 481–489, 2002) have recently shown that the weighted version of this problem is at least as hard as set cover; they have also developed a 3-approximation algorithm for the unweighted version. We give a 2-approximation algorithm for the unweighted version, improving the Chuzhoy-Naor bound of 3 and matching (up to lower-order terms) the best approximation ratio known for the vertex cover problem.

Keywords and Phrases

Approximation algorithms capacitated covering set cover vertex cover linear programming randomized rounding 

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References

  1. 1.
    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
  2. 2.
    J. Bar-Ilan, G. Kortsarz and D. Peleg. Generalized submodular cover problems and applications. Theoretical Computer Science, 250, pages 179–200, 2001.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    V. Chvátal. A Greedy Heuristic for the Set Covering Problem. Mathematics of Operations Research, vol. 4, No 3, pages 233–235, 1979.MATHMathSciNetGoogle Scholar
  4. 4.
    R. D. Carr, L. K. Fleischer, V. J. Leung and C. A. Phillips. Strengthening Integrality Gaps For Capacitated Network Design and Covering Problems. In Proc. of the 11th ACM-SIAM Symposium on Discrete Algorithms, pages 106–115, 2000.Google Scholar
  5. 5.
    J. Chuzhoy and J. Naor. Covering Problems with Hard Capacities. Proc. of the Forty-Third IEEE Symp. on Foundations of Computer Science, 481–489, 2002.Google Scholar
  6. 6.
    G. Dobson. Worst Case Analysis of Greedy Heuristics For Integer Programming with Non-Negative Data. Math. of Operations Research, 7(4):515–531, 1980.MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Gandhi, S. Khuller, S. Parthasarathy and A. Srinivasan. Dependent Rounding in Bipartite Graphs. In Proc. of the Forty-Third IEEE Symposium on Foundations of Computer Science, pages 323–332, 2002.Google Scholar
  8. 8.
    S. Guha, R. Hassin, S. Khuller and E. Or. Capacitated Vertex Covering with Applications. Proc. ACM-SIAM Symp. on Discrete Algorithms, pages 858–865, 2002.Google Scholar
  9. 9.
    D. S. Johnson, Approximation algorithms for combinatorial problems. J. Computer and System Sciences, 9, pages 256–278, 1974.MATHCrossRefGoogle Scholar
  10. 10.
    E. Halperin. Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, California, pages 329–337, 2000.Google Scholar
  11. 11.
    D. S. Hochbaum. Approximation Algorithms for the Set Covering and Vertex Cover Problems. SIAM Journal on Computing, 11:555–556, 1982.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D. S. Hochbaum. Heuristics for the fixed cost median problem. Mathematical Programming, 22:148–162, 1982.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    D. S. Hochbaum (editor). Approximation Algorithms for NP-hard Problems. PWS Publishing Company, 1996.Google Scholar
  14. 14.
    S. G. Kolliopoulos and N. E. Young. Tight Approximation Results for General Covering Integer Programs. In Proc. of the Forty-Second Annual Symposium on Foundations of Computer Science, pages 522–528, 2001.Google Scholar
  15. 15.
    L. Lovász, On the ratio of optimal integral and fractional covers. Discrete Math., 13, pages 383–390, 1975.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Pál, É. Tardos and T. Wexler. Facility Location with Nonuniform Hard Capacities. In Proc. Forty-Second Annual Symposium on Foundations of Computer Science, 329–338, 2001.Google Scholar
  17. 17.
    V. Vazirani. Approximation Algorithms. Springer-Verlag, 2001.Google Scholar
  18. 18.
    L. A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2:385–393, 1982.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    N. E. Young. K-medians, facility location, and the Chernoff-Wald bound. ACM-SIAM Symposium on Discrete Algorithms, pages 86–95, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rajiv Gandhi
    • 1
  • Eran Halperin
    • 2
    • 6
  • Samir Khuller
    • 3
  • Guy Kortsarz
    • 4
  • Aravind Srinivasan
    • 5
  1. 1.Department of Computer ScienceUniversity of MarylandCollege Park
  2. 2.International Computer Science InstituteBerkeley
  3. 3.Department of Computer Science and Institute for Advanced Computer StudiesUniversity of MarylandCollege Park
  4. 4.Department of Computer ScienceRutgers UniversityCamden
  5. 5.Department of Computer Science and Institute for Advanced Computer StudiesUniversity of MarylandCollege Park
  6. 6.Computer Science DivisionUniversity of CaliforniaBerkeley

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