NP-Hard Classes of Linear Algebraic Systems with Uncertainties
- 25 Downloads
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) = ∑CiXi) 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.
Unable to display preview. Download preview PDF.
- 1.Beckenbach, E. F. and Bellman, R.: Inequalities, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1961.Google Scholar
- 2.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.Google Scholar
- 3.Garey, M. R. and Johnson, D. S.: Computers and Intractability: a Guide to the Theory of NPCompleteness, W. F. Freeman, San Francisco, 1979.Google Scholar
- 4.Khachiyan, L. G.: A Polynomial-Time Algorithm for Linear Programming, Soviet Math. Dokl. 20 (1) (1979), pp. 191–194.Google Scholar
- 5.Kolmogorov, A. N. and Fomin, S. V.: Elements of the Theory of Functions and Functional Analysis, Nauka Publ., Moscow, 1972 (in Russian).Google Scholar
- 6.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.Google Scholar
- 7.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.Google Scholar
- 8.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.Google Scholar
- 9.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).Google Scholar
- 10.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.Google Scholar
- 11.Poljak, S. and Rohn, J.: Checking Robust Non-Singularity is NP-Hard,Mathematics of Control, Signals and Systems 6 (1993), pp. 1–9.Google Scholar
- 12.Rohn, J.: Systems of Linear Interval Equations, Linear Algebra and its Applications 126 (1989), pp. 39–78.Google Scholar
- 13.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.Google Scholar
- 14.Shary, S. P.: Optimal Solution of Interval Linear Algebraic Systems. I, Interval Computations 2 (1991), pp. 7–30.Google Scholar
- 15.Shary, S. P.:A New Class of Algorithms forOptimal Solution of Interval Linear Systems, Interval Computations 2 (4) (1992), pp. 18–29.Google Scholar
- 16.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.Google Scholar