Skip to main content
SpringerLink
Log in
Book cover

Annual Symposium on Theoretical Aspects of Computer Science

STACS 1994: STACS 94 pp 521–532Cite as

On the approximability of finding maximum feasible subsystems of linear systems

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 775))

Abstract

We consider the combinatorial problem MAXFLS which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The approximability of this general problem is investigated for the three types of relations =, ≥ and >. Various constrained versions of MAXFLS where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We show that MAXFLS with =, ≥ or > relations is NP-hard even when restricted to homogeneous systems with bipolar coefficients. The various NP-hard versions of MAXFLS belong to different approximability classes depending on the type of relations and the additional constraints. While MAXFLS with equations and integer coefficients cannot be approximated within ε for some ε > 0 where p is the number of relations, MAXFLS with strict or nonstrict inequalities can be approximated within 2 but not within every constant factor.

supported by grants from TFR

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Amaldi. On the complexity of training perceptions. In T. Kohonen et al., editor, Artificial Neural Networks, pages 55–60, Amsterdam, 1991. Elsevier science publishing company.

    Google Scholar 

  2. E. Amaldi and V. Kann, 1993. Manuscript in preparation.

    Google Scholar 

  3. E. Amaldi and V. Kann. The complexity and approximability of finding maximum feasible subsystems of linear relations. Technical Report ORWP-11-93, Department of Mathematics, Swiss Federal Institute of Technology, Lausanne and Technical Report TRITA-NA-9313, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm, 1993.

    Google Scholar 

  4. S. Arora, L. Babai, J. Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equation. In Proc. of 34rd Ann. IEEE Symp. on Foundations of Comput. Sci., pages 724–733, 1993.

    Google Scholar 

  5. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems. In Proc. of 33rd Ann. IEEE Symp. on Foundations of Comput. Sci., pages 14–23, 1992.

    Google Scholar 

  6. P. Berman and G. Schnitger. On the complexity of approximating the independent set problem. Inform. and Comput., 96:77–94, 1992.

    Google Scholar 

  7. M. R. Garey and D. S. Johnson. Computers and Intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company, San Francisco, 1979.

    Google Scholar 

  8. R. Greer. Trees and Hills: Methodology for Maximizing Functions of Systems of Linear Relations, volume 22 of Annals of Discrete Mathematics. Elsevier science publishing company, Amsterdam, 1984.

    Google Scholar 

  9. J. Håstad, S. Phillips, and S. Safra. A well-characterized approximation problem. Inform. Process. Lett., 47:301–305, 1993.

    Google Scholar 

  10. K-U. Höffgen, H-U. Simon, and K. van Horn. Robust trainability of single neurons. Technical Report CS-92-9, Computer Science Department, Brigham Young University, Provo, 1992.

    Google Scholar 

  11. D. S. Johnson and F. P. Preparata. The densest hemisphere problem. Theoretical Computer Science, 6:93–107, 1978.

    Google Scholar 

  12. V. Kann. On the Approximability of NP-complete Optimization Problems. PhD thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm, 1992.

    Google Scholar 

  13. N. Karmarkar. A new polynomial time algorithm for linear programming. Combinatorica, 4:373–395, 1984.

    Google Scholar 

  14. C. Lund and M. Yannakakis. On the hardness of approximating minimization problems. In Proc. Twenty fifth Ann. ACM Symp. on Theory of Comp., pages 286–293, 1993.

    Google Scholar 

  15. C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comput. System Sci., 43:425–440, 1991.

    Google Scholar 

  16. R. E. Warmack and R. C. Gonzalez. An algorithm for optimal solution of linear inequalities and its application to pattern recognition. IEEE Trans. on Computers, 22:1065–1075, 1973.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dep. of Mathematics, Swiss Federal Institute of Technology, CH-1015, Lausanne

    Edoardo Amaldi

  2. Dep. of Numerical Analysis and Computing Science, Royal Institute of Technology, S-100 44, Stockholm

    Viggo Kann

Authors
  1. Edoardo Amaldi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Viggo Kann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Patrice Enjalbert Ernst W. Mayr Klaus W. Wagner

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Amaldi, E., Kann, V. (1994). On the approximability of finding maximum feasible subsystems of linear systems. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_168

Download citation

  • DOI: https://doi.org/10.1007/3-540-57785-8_168

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57785-0

  • Online ISBN: 978-3-540-48332-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation