Advertisement

P-Completeness of Testing Solutions of Parametric Interval Linear Systems

  • Milan HladíkEmail author
Chapter
  • 7 Downloads
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 276)

Abstract

We deal with a system of parametric interval linear equations and also with its particular sub-classes defined by symmetry of the constraint matrix. We show that the problem of checking whether a given vector is a solution is a P-complete problem, meaning that there unlikely exists a polynomial closed form arithmetic formula describing the solution set. This is true not only for the general parametric system, but also for the symmetric case with general linear dependencies in the right-hand side. However, we leave as an open problem whether P-completeness concerns also the simplest version of the symmetric solution set with no dependencies in the right-hand side interval vector.

Notes

Acknowledgements

The author was supported by the Czech Science Foundation Grant P403-18-04735S.

References

  1. 1.
    Alefeld, G., Kreinovich, V., Mayer, G.: On the shape of the symmetric, persymmetric, and skew-symmetric solution set. SIAM J. Matrix Anal. Appl. 18(3), 693–705 (1997)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alefeld, G., Kreinovich, V., Mayer, G.: On the solution sets of particular classes of linear interval systems. J. Comput. Appl. Math. 152(1–2), 1–15 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alefeld, G., Mayer, G.: On the symmetric and unsymmetric solution set of interval systems. SIAM J. Matrix Anal. Appl. 16(4), 1223–1240 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Goldschlager, L.M., Shaw, R.A., Staples, J.: The maximum flow problem is log space complete for P. Theor. Comput. Sci. 21, 105–111 (1982)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, New York (1995)Google Scholar
  6. 6.
    Hladík, M.: Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl. 30(2), 509–521 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hladík, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561–574 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lueker, G.S., Megiddo, N., Ramachandran, V.: Linear programming with two variables per inequality in poly-log time. In: Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, STOC ’86, pp. 196–205. ACM, New York (1986)Google Scholar
  9. 9.
    Mayer, G.: An Oettli-Prager-like theorem for the symmetric solution set and for related solution sets. SIAM J. Matrix Anal. Appl. 33(3), 979–999 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mayer, G.: A survey on properties and algorithms for the symmetric solution set. Technical Report 12/2, Universität Rostock, Institut für Mathematik (2012). http://ftp.math.uni-rostock.de/pub/preprint/2012/pre12_02.pdf
  11. 11.
    Mayer, G.: Three short descriptions of the symmetric and of the skew-symmetric solution set. Linear Algebra Appl. 475, 73–79 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Oettli, W., Prager, W.: Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math. 6, 405–409 (1964)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Popova, E.D.: Explicit description of \(AE\) solution sets for parametric linear systems. SIAM J. Matrix Anal. Appl. 33(4), 1172–1189 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Popova, E.D.: Solvability of parametric interval linear systems of equations and inequalities. SIAM J. Matrix Anal. Appl. 36(2), 615–633 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Schrijver, A.: Theory of Linear and Integer Programming. Repr. Wiley, Chichester (1998)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Mathematics and Physics, Department of Applied MathematicsCharles UniversityPragueCzech Republic

Personalised recommendations