Abstract
For a system of linear equations Ax = b, the following natural questions appear:
• does this system have a solution?
• if it does, what are the possible values of a given objective function f(x1,...,xn) (e.g., of a linear function f(x) = ∑C i X i ) over the system's solution set?
We show that for several classes of linear equations with uncertainty (including interval linear equations) these problems are NP-hard. In particular, we show that these problems are NP-hard even if we consider only systems of n+2 equations with n variables, that have integer positive coefficients and finitely many solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beckenbach, E. F. and Bellman, R.: Inequalities, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1961.
Cormen, Th. H., Leiserson, C. E., and Rivest, R. L.: Introduction to Algorithms, MIT Press, Cambridge, MA, and Mc-Graw Hill Co., N.Y., 1990.
Garey, M. R. and Johnson, D. S.: Computers and Intractability: a Guide to the Theory of NPCompleteness, W. F. Freeman, San Francisco, 1979.
Khachiyan, L. G.: A Polynomial-Time Algorithm for Linear Programming, Soviet Math. Dokl. 20 (1) (1979), pp. 191–194.
Kolmogorov, A. N. and Fomin, S. V.: Elements of the Theory of Functions and Functional Analysis, Nauka Publ., Moscow, 1972 (in Russian).
Kreinovich, V., Lakeyev, A. V., and Noskov, S. I.: Optimal Solution of Interval Linear Systems is Intractable (NP-Hard), Interval Computations 1 (1993), pp. 6–14.
Kreinovich, V., Lakeyev, A. V., and Noskov, S. I.: Approximate Linear Algebra is Intractable, Linear Algebra and its Applications 232 (1) (1996), pp. 45–54.
Lakeyev, A. V. and Noskov, S. I.: A Description of the Set of Solutions of a Linear Equation with Interval Defined Operator and Right-Hand Side, Russian Acad. Sci. Dokl. Math. 47 (3) (1993), pp. 518–523.
Lakeyev, A.V. and Noskov, S. I.:On the Set of Solutions of the Linear Equationwith the Intervally Given Operator and the Right-Hand Side, Siberian Math. J. 35 (5) (1994), pp. 1074–1084 (in Russian).
Oettli, W. and Prager, W.: Compatibility of Approximate Solution of Linear Equations with Given Error Bounds for Coefficients and Right-Hand Sides, Num. Math. 6 (1964), pp. 405–409.
Poljak, S. and Rohn, J.: Checking Robust Non-Singularity is NP-Hard,Mathematics of Control, Signals and Systems 6 (1993), pp. 1–9.
Rohn, J.: Systems of Linear Interval Equations, Linear Algebra and its Applications 126 (1989), pp. 39–78.
Rohn, J. and Kreinovich, V.: Computing Exact Componentwise Bounds on Solutions of Linear Systems with Interval Data is NP-Hard, SIAM J. Matrix Anal. Appl. 16 (2) (1995), pp. 415–420.
Shary, S. P.: Optimal Solution of Interval Linear Algebraic Systems. I, Interval Computations 2 (1991), pp. 7–30.
Shary, S. P.:A New Class of Algorithms forOptimal Solution of Interval Linear Systems, Interval Computations 2 (4) (1992), pp. 18–29.
Shary, S. P.: Solving Interval Linear Systems with Nonnegative Matrices, in: Markov, S. M. (ed.), Proc. of the Conference “Scientific Computation and Mathematical Modelling”, DATECS Publishing, Sofia, 1993, 179–181.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lakeyev, A.V., Kreinovich, V. NP-Hard Classes of Linear Algebraic Systems with Uncertainties. Reliable Computing 3, 51–81 (1997). https://doi.org/10.1023/A:1009938325229
Issue Date:
DOI: https://doi.org/10.1023/A:1009938325229