Skip to main content
SpringerLink
Log in

Probabilistic combinatorial optimization

  • Communications
  • Conference paper
  • First Online:
Mathematical Foundations of Computer Science 1981 (MFCS 1981)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 118))

Abstract

The (bounded) generalized maximum satisfiability problem covers a broad range of NP-complete problems, e.g. it is a generalization of INDEPENDENT SET, LINEAR INEQUALITY, HITTING SET, SET PACKING, MINIMUM COVER, etc. The complexity of finding approximations for problems in this class is analyzed. The results have several interpretations, including the following:

  • - A general class of existence proofs is made efficiently constructive.

  • - A class of randomized algorithms is made deterministic and efficient.

  • - A new class of combinatorial approximation algorithms is introduced, which is based on “background” optimization. Instead of maximizing among all assignments we maximize among expected values for parametrized random solutions. It turns out that this “background” optimization is in two precise senses best possible if P≠NP. The “background optimization” performed is equivalent to finding the maximum of a polynomial in a bounded region.

This research is supported by National Science Foundation grants MCS80-04490 and ENG76-16808.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Erdos and J. Spencer, Probabilistic methods in combinatorics, Academic Press, New York (1974).

    Google Scholar 

  2. P. Erdos and D.J. Kleitman, “On coloring graphs to maximize the proportion of multicolored k-edges,” Journal of Combinatorial Theory 5(2), pp.164–169 (Sept. 1968).

    Google Scholar 

  3. K. Lieberherr, “Polynomial and absolute P-optimal algorithms for a relaxation of generalized maximum satisfiability,” Report 276, Dep. of EECS, Princeton University (1980).

    Google Scholar 

  4. K. Lieberherr and E. Specker, “Complexity of partial satisfaction,” Journal of the ACM 28 (1981).

    Google Scholar 

  5. L. Lovasz, Combinatorial problems and exercises, North-Holland Publishing Company, New York (1979).

    Google Scholar 

  6. T. Schaefer, “The complexity of satisfiability problems,” Proc. 10th Annual ACM Symposium on Theory of Computing, pp.216–226 (1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Princeton University, 08544, Princeton, New Jersey

    Karl Lieberherr

Authors
  1. Karl Lieberherr
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jozef Gruska Michal Chytil

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lieberherr, K. (1981). Probabilistic combinatorial optimization. In: Gruska, J., Chytil, M. (eds) Mathematical Foundations of Computer Science 1981. MFCS 1981. Lecture Notes in Computer Science, vol 118. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-10856-4_110

Download citation

  • DOI: https://doi.org/10.1007/3-540-10856-4_110

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10856-6

  • Online ISBN: 978-3-540-38769-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation