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.
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References
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)
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)
Alefeld, G., Mayer, G.: On the symmetric and unsymmetric solution set of interval systems. SIAM J. Matrix Anal. Appl. 16(4), 1223–1240 (1995)
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)
Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory. Oxford University Press, New York (1995)
Hladík, M.: Description of symmetric and skew-symmetric solution set. SIAM J. Matrix Anal. Appl. 30(2), 509–521 (2008)
Hladík, M.: Enclosures for the solution set of parametric interval linear systems. Int. J. Appl. Math. Comput. Sci. 22(3), 561–574 (2012)
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)
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)
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
Mayer, G.: Three short descriptions of the symmetric and of the skew-symmetric solution set. Linear Algebra Appl. 475, 73–79 (2015)
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)
Popova, E.D.: Explicit description of \(AE\) solution sets for parametric linear systems. SIAM J. Matrix Anal. Appl. 33(4), 1172–1189 (2012)
Popova, E.D.: Solvability of parametric interval linear systems of equations and inequalities. SIAM J. Matrix Anal. Appl. 36(2), 615–633 (2015)
Schrijver, A.: Theory of Linear and Integer Programming. Repr. Wiley, Chichester (1998)
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The author was supported by the Czech Science Foundation Grant P403-18-04735S.
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Hladík, M. (2020). P-Completeness of Testing Solutions of Parametric Interval Linear Systems. In: Ceberio, M., Kreinovich, V. (eds) Decision Making under Constraints. Studies in Systems, Decision and Control, vol 276. Springer, Cham. https://doi.org/10.1007/978-3-030-40814-5_14
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DOI: https://doi.org/10.1007/978-3-030-40814-5_14
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