Approximation Hardness for Small Occurrence Instances of NP-Hard Problems

  • Miroslav Chlebík
  • Janka Chlebíková
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2653)


The paper contributes to the systematic study (started by Berman and Karpinski) of explicit approximability lower bounds for small occurrence optimization problems.We present parametrized reductions for some packing and covering problems, including 3-Dimensional Matching, and prove the best known inapproximability results even for highly restricted versions of them. For example, we show that it is NP-hard to approximate Max-3-DM within \( \frac{{139}} {{138}} \) even on instances with exactly two occurrences of each element. Previous known hardness results for bounded occurence case of the problem required that the bound is at least three, and even then no explicit lower bound was known.

New structural results which improve the known bounds for 3-regular amplifiers and hence the inapproximability results for numerous small occurrence problems studied earlier by Berman and Karpinski are also presented.


Hardness Result Steiner Tree Problem Contact Node Inapproximability Result Small Occurrence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Berman and T. Fujito: Approximating independent sets in degree 3 graphs, Proc. 4th Workshop on Algorithms and Data Structures, 1995, Springer-Verlag, Berlin, LNCS 955, 449–460.Google Scholar
  2. 2.
    P. Berman and M. Karpinski: On some tighter inapproximability results, further improvements, ECCC, Report No. 65, 1998.Google Scholar
  3. 3.
    L. Engebretsen and M. Karpinski: Approximation hardness of TSP with bounded metrics, Proceedings of 28th ICALP, 2001, LNCS 2076, 201–212.Google Scholar
  4. 4.
    J. Håstad: Some optimal inapproximability results, Journal of ACM 48 (2001), 798–859.CrossRefzbMATHGoogle Scholar
  5. 5.
    M. M. Halldórsson: Approximating k-set cover and complementary graph coloring, Proc. 5th International Conference on Integer Programming and Combinatorial Optimization, 1996, Springer-Verlag, Berlin, LNCS 1084, 118–131.Google Scholar
  6. 6.
    C. A. J. Hurkens and A. Schrijver: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Mathematics 2 (1989), 68–72.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    M. Chlebík and J. Chlebíková: Approximation Hardness of the Steiner Tree Problem on Graphs, Proceedings of the 8th Scandinavian Workshop on Algorithm Theory, SWAT 2002, Springer, LNCS 2368, 170–179.Google Scholar
  8. 8.
    M. Chlebík and J. Chlebíková: Approximation Hardness for Small Occurrence Instances of NP-Hard Problems, ECCC, Report No. 73, 2002.Google Scholar
  9. 9.
    V. Kann: Maximum bounded 3-dimensional matching is MAX SNP-complete, Information Processing Letters 37 (1991), 27–35.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    C. H. Papadimitriou and M. Yannakakis: Optimization, approximation, and complexity classes, J. Computer and System Sciences 43 (1991), 425–440.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    C. H. Papadimitriou and S. Vempala: On the Approximability of the Traveling Salesman Problem, Proceedings of the 32nd ACM Symposium on the theory of computing, Portland, 2000.Google Scholar
  12. 12.
    M. Thimm: On the Approximability of the Steiner Tree Problem, Proceedings of the 26th International Symposium, MFCS 2001, Springer, LNCS 2136, 678–689.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Miroslav Chlebík
    • 1
  • Janka Chlebíková
    • 2
  1. 1.MPI for Mathematics in the SciencesLeipzig
  2. 2.Institut für Informatik und Praktische MathematikCAUKiel

Personalised recommendations