Abstract
In many real-life applications of interval computations, the desired quantities appear (in a good approximation to reality) as a solution to a system of interval linear equations. It is known that such systems are difficult to solve (NP-hard) but still algorithmically solvable. If instead of the (approximate) interval linear systems, we consider more realistic (and more general) formulations, will the corresponding problems still be algorithmically solvable? We consider three natural generalizations of interval linear systems: to conditions which are more general than linear systems, to multi-intervals instead of intervals, and to dynamics (differential and difference equations) instead of statics (linear and algebraic equations). We show that the problem is still algorithmically solvable for non-linear systems and even for more general conditions, and it is still solvable if we consider linear or non-linear systems with multi-intervals instead of intervals. However, generalized conditions with multi-interval uncertainty are already algorithmically unsolvable. For dynamics: difference equations are still algorithmically solvable, differential equations are, in general, unsolvable.
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References
Th. H. Cormen, C. E. Leiserson, and R. L. Rivest (1990), Introduction to algorithms, MIT Press, Cambridge, MA, and Mc-Graw Hill Co., N.Y.
M. Davis, Yu. V. Matiyasevich, and J. Robinson (1976), “Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution”, In: Mathematical developments arising from Hilbert’s problems, Proceedings of Symposia in Pure Mathematics, Vol. 28, American Math. Society, Providence, RI, Part 2, pp. 323–378.
M. E. Garey and D. S. Johnson (1979), Computers and intractability: a guide to the theory of NP-completeness, Freeman, San Francisco.
J. Hastad (1997), “Some optimal inapproximability results”, Proceedings of the Annual ACM Symposium on Theory of Computing STOC’97, El Paso, TX, May 4–6, 1997, ACM Press, pp. 1–10.
D. Hilbert (1902), “Mathematical Problems” (lecture delivered before the International Congress of Mathematics in Paris in 1900), translated in Bull. Amer. Math, Soc., Vol. 8, pp. 437–479; reprinted in Mathematical developments arising from Hilbert’s problems, Proceedings of Symposia in Pure Mathematics, Vol. 28, American Math. Society, Providence, RI, 1976, Part 1, pp. 1–34.
V. Kreinovich, A. V. Lakeyev, and S. I. Noskov (1993), “Optimal solution of interval linear systems is intractable (NP-hard).” Interval Computations, No. 1, pp. 6–14.
V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl (1997), Computational complexity and feasibility of data processing and interval computations, Kluwer, Dordrecht.
A. V. Lakeyev and S. I. Noskov (1993), “A description of the set of solutions of a linear equation with interval defined operator and right-hand side” Russian Academy of Sciences, Doklady, Mathematics, Vol. 47, No. 3, pp. 518–523.
A. V. Lakeyev and S. I. Noskov (1994), “On the solution set of a linear equation with the right-hand side and operator given by intervals”, Siberian Math. J., Vol. 35, No. 5, pp. 957–966.
Yu. V. Matiyasevich (1970), “Enumerable sets are diophantine”, Soviet Math. Doklady, Vol. 11, pp. 354–357.
Yu. V. Matiyasevich and J. Robinson (1974), “Reduction of an arbitrary Diophantine equation to one in 13 unknowns”, Acta Arithmetica, Vol. 27, pp. 521553.
A. Seidenberg (1954), “A new decision method for elementary algebra”, Annals of Math., Vol. 60, pp. 365–374.
S. P. Shary (1996), “Algebraic approach to the interval linear static identification, tolerance, and control problems, or One more application of Kaucher arithmetic”, Reliable Computing, Vol. 2, No. 1, pp. 3–34.
A. Tarski (1951), A decision method for elementary algebra and geometry,2nd ed., Berkeley and Los Angeles.
U. Zwick (1998), “Finding almost-satisfying assignments”, Proceedings of the Annual ACM Symposium on Theory of Computing STOC’98, Dallas, TX, May 23–26, 1998, ACM Press, pp. 551–560.
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Kreinovich, V. (2000). Beyond Interval Systems: What Is Feasible and What Is Algorithmically Solvable?. In: Pardalos, P.M. (eds) Approximation and Complexity in Numerical Optimization. Nonconvex Optimization and Its Applications, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3145-3_22
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DOI: https://doi.org/10.1007/978-1-4757-3145-3_22
Publisher Name: Springer, Boston, MA
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