Advertisement

Approximation of a Geometric Set Covering Problem

  • Sofia Kovaleva
  • Frits C. R. Spieksma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2223)

Abstract

We consider a special set covering problem. This problem is a generalization of finding a minimum clique cover in an interval graph. When formulated as an integer program, the 0-1 constraint matrix of this integer program can be partitioned into an interval matrix and a special 0-1 matrix with a single 1 per column. We show that the value of this formulation is bounded by \( \tfrac{{2k}} {{k + 1}} \) times the value of the LP-relaxation, where k is the maximum row sum of the special matrix. For the “smallest” difficult case, i.e., k = 2, this bound is tight. Also we provide an O(n) 3/2 -approximation algorithm in case k = 2.

Keywords

Integer Program Covering Problem Interval Graph Constraint Matrix Interval Matrix 
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.
    Arora, S., C. Lund, R. Motwani, M. Sudan, and M. Szegedy (1992), ‘Proof verification and hardness of approximation problems”, Proceedings of the 33rd IEEE Symposium on the Foundations of Computer Science, 14–23.Google Scholar
  2. 2.
    Bar-Noy, A., S. Gupta, J. Naor and B. Schieber (1999), “Approximating the Throughput of Multiple Machines in Real-Time Scheduling”, Proceedings of the 31st ACM Symposium on Theory of Computing.Google Scholar
  3. 3.
    Garg, N., V. V. Vazirani and M. Yannakakis (1997), “Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees”, Algorithmica 18, 3–20.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Golumbic, M.C, Algorithmic Graph Theory and Perfect Graphs, Academic Press, San Diego, California, 1980.MATHGoogle Scholar
  5. 5.
    Hochbaum, D. S. (1998), “Instant Recognition of Half Integrality and 2-Approximations”, in: Proceedings of the APPROX’98 Conference, editors: Klaus Jansen and José Rolim, Lecture Notes in Computer Science 1444, 99–110.Google Scholar
  6. 6.
    Schrijver, A., Theory of Linear and Integer Programming, John Wiley & Sons, 1998.Google Scholar
  7. 7.
    Aerts, J., Marinissen, E. J. (1998), “Scan chain design for test time reduction in core-based ICs”, Paper 18.1 at International test conference, Washington, D. C., October 1998.Google Scholar
  8. 8.
    Kovaleva, S., Spieksma, F. C. R. (2001), “Primal-dual approximation algorithms for a packing-covering pair of problems”, Report M 01-01, Department of Mathematics, Maastricht University, P. O.Box 616, NL-6200 MD, Maastricht, The Netherlands.Google Scholar
  9. 9.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M., Complexity and Approximation, Springer-Verlag, Berlin, 1999.Google Scholar
  10. 10.
    Nemhauser, G. L., Wolsey, L. A., Integer and Combinatorial Optimization, Wiley & Sons, 1999.Google Scholar
  11. 11.
    Hoffman, A. J., Kolen, A. W. J., Sakarovitch, M., (1985), “Totally-balanced and greedy matrices”, SIAM Journal of Algebraic and Discrete Methods 6, 721–730.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Sofia Kovaleva
    • 1
  • Frits C. R. Spieksma
    • 1
  1. 1.Department of MathematicsMaastricht UniversityMaastrichtThe Netherlands

Personalised recommendations