On a Refined Analysis of Some Problems in Interval Arithmetic Using Real Number Complexity Theory
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We study some problems in interval arithmetic treated in Kreinovich et al. . First, we consider the best linear approximation of a quadratic interval function. Whereas this problem (as decision problem) is known to be NP-hard in the Turing model, we analyze its complexity in the real number model and the analogous class NPℝ. Our results substantiate that most likely it does not any longer capture the difficulty of NPℝ in such a real number setting. More precisely, we give upper complexity bounds for the approximation problem for interval functions by locating it in (a real analogue of). This result allows several conclusions:
• the problem is not (any more) NPℝ-hard under so called weak polynomial time reductions and likely not to be NPℝ-hard under (full) polynomial time reductions;
• for fixed dimension the problem is polynomial time solvable; this extends the results in Koshelev et al.  and answers a question left open in .
We also study several versions of interval linear systems and show similar results as for the approximation problem.
Our methods combine structural complexity theory with issues from semi-infinite optimization theory.
KeywordsApproximation Problem Interval Function Complexity Theory Analogous Class Interval Arithmetic
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- 1.Blum, L., Cucker, F., Shub, M., and Smale, S.: Complexity and Real Computation, Springer, 1998.Google Scholar
- 2.Bochnak, J., Coste, M., and Roy, M. F.: Real Algebraic Geometry, Springer, 1998.Google Scholar
- 3.Cucker, F. and Matamala, M.: On Digital Nondeterminism, Mathematical Systems Theory 29 (1996), pp. 635–647.Google Scholar
- 4.Cucker, F., Shub, M., and Smale, S.: Complexity Separations in Koiran's Weak Model, Theoretical Computer Science 133 (1994), pp. 3–14.Google Scholar
- 5.Gelfand, I. M., Kapranov, M. M., and Zelevinsky, A. V.: Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, 1994.Google Scholar
- 6.Grädel, E. and Meer, K.: Descriptive Complexity Theory over the Real Numbers, in: Renegar, J., Shub, M., and Smale, S. (eds), Lectures in Applied Mathematics 32, 1996, pp. 381–403.Google Scholar
- 7.Gustafson, S. Å. and Kortanek, K. O.: Semi-Infinte Programming and Applications, in: Bachem, A., Grötschel, M., and Korte, B. (eds), Mathematical Programming: The State of the Art, Springer, 1983, pp. 132–157.Google Scholar
- 8.Hettich, R., and Jongen, H. Th.: Semi-Infinite Programming: Conditions of Optimality and Applications, in: Stoer, J. (ed.), Optimization Techniques, Part 2, Lecture Notes in Control and Inform. Sci. 7, Springer, 1978, pp. 1–11.Google Scholar
- 9.Hettich, R. and Kortanek, K. O.: Semi-Infinite Programming, SIAM Review 35(3) (1993), pp. 380–429.Google Scholar
- 10.Jongen, H. Th., Twilt, F., and Weber, G. W.: Semi-Infinite Optimization: Structure and Stability of the Feasible Set, Journal Opt. Theory Applications 72 (1992), pp. 529–552.Google Scholar
- 11.Koiran, P.: A Weak Version of the Blum-Shub-Smale Model, in: 34th Annual IEEE Symposium on Foundations of Computer Science, 1993, pp. 486–495.Google Scholar
- 12.Koshelev, M., Longpré, L., and Taillibert, P.: Optimal Enclusure of Quadratic Interval Functions, Reliable Computing 4(4) (1998), pp. 351–360.Google Scholar
- 13.Kreinovich, V., Lakeyev, A. V., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, 1997.Google Scholar
- 14.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, Russain Acad. Sci. Dokl. Math. 47(3) (1993), pp. 518–523.Google Scholar
- 15.Meer, K.: On the Complexity of Quadratic Programming in Real Number Models of Computation, Theoretical Computer Science 133 (1994), pp. 85–94.Google Scholar
- 16.Mumford, D.: Algebraic Geometry I: Complex Projective Varieties, Springer, 1976.Google Scholar
- 17.Plaisted, D. A.: New NP-Hard and NP-Complete Polynomial and Integer Divisibility Problems, Theoretical Computer Science 31 (1984), pp. 125–138.Google Scholar
- 18.Shub, M.: Some Remarks on Bezout's Theorem and Complexity Theory, in: Hirsch, M., Marsden, J., and Shub, M. (eds), Form Topology to Computation: Proc. of the Smalefest, Springer, 1993, pp. 443–455.Google Scholar
- 19.Stockmeyer, L.: The Polynomial-Time Hierarchy, Theoretical Computer Science 3 (1977), pp. 1–22.Google Scholar
- 20.Sturmfels, B.: Introduction to Resultants, in: Cox, D. A. and Sturmfels, B. (eds), Application of Computational Algebraic Geometry, Proc. of Symposia in Applied Mathematics 53, American Mathematical Society, 1998, pp. 25–39.Google Scholar
- 21.Tichatschke, R., Hettich, R., and Still, G.: Connections between Generalized, Inexact and Semi-Infinite Linear Programming, ZOR—Methods and Models of Operations Research 33 (1989), pp. 367–382.Google Scholar
- 22.Woźniakowski, H.: Why Does Information-Based Complexity Use the Real Number Model? Theoretical Computer Science 219 (1999), pp. 451–465.Google Scholar