Reliable Computing

, Volume 1, Issue 3, pp 343–359 | Cite as

A bright side of NP-hardness of interval computations: interval heuristics applied to NP-problems

  • Bonnie Traylor
  • Vladik Kreinovich
Mathematical Research


It is known that interval computations are NP-hard. In other words, the solution of many important problems can be reduced to interval computations. The immediate conclusion is negative: in the general case, one cannot expect an algorithm to do all the interval computations in less than exponential running time.

We show that this result also has a bright side: since there are many heuristics, for interval computations, we can solve other problems by reducing them to interval computations and applying these heuristics.


Mathematical Modeling Computational Mathematic Industrial Mathematic Interval Computation Bright Side 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Въічодная сторона ПР-сложности интервальных вычислений: интервалъная зврицтика в применении К ПР-задачам


Извецтно, что интервалъные вычиления ПР-сложны. Друтими словами, решение многх важных задач может быжтъ сведено к интервалъным вычицлениям. Первое очевидное следствие зтого Факта негативно: в обшем слмчае мы не можем поцтроитъ алгстроитм, который выполнял цы все интервалъные вычисления быстрее, чем за зксноненциалън ое время.

Нами показано, что зто свойство имеет и свою выгодную сторону: посколъку для интервалъных выцислений сушествует мното звристик, другие задачи могут бытъ вешены решены сведением их к интервалъным вычислениям с далънейшим применением зтих звристик.


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  1. [1]
    Adams, D.Life the Universe and everything. Pocket Books, N.Y., 1983.Google Scholar
  2. [2]
    Dubois, O.,Counting the number of solutions for instances of satisfiability. Theoretical Computer Science81 (1991), pp. 49–64.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Gaganov, A. A.Computational complexity of the range of the polynomial in several variables. Cybernetics (1985), pp. 418–421.Google Scholar
  4. [4]
    Garey, M. and Johnson, D.Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, 1979.Google Scholar
  5. [5]
    Hansen, E. R.On solving systems of equations using interval arithmetic. Math. Comp. (1968), pp. 374–384.Google Scholar
  6. [6]
    Kolacz, H.On the optimality of inclusion algorithms. In: Nickel, K. (ed.) “Interval Mathematics 1985”, Lecture Notes in Computer Science 212, Springer-Verlag, Berlin, Heidelberg, N.Y., 1986, pp. 67–79.Google Scholar
  7. [7]
    Koutsoupias, E. and Papadimitriou, C. H.On the greedy algorithm for satisfiability. Information Processing Letters43 (1992), pp. 53–55.CrossRefMathSciNetGoogle Scholar
  8. [8]
    Kreinovich, V.Maslov's increasing freedom of choice strategy: on the example of the satisfiability problem. Leningrad Center for New Informational Technology “Informatika”, Technical Report, 1989 (in Russian).Google Scholar
  9. [9]
    Kreinovich, V., Bernat, A., Villa, E., and Mariscal, Y.Parallel computers estimate errors caused by imprecise data. Interval Computations, 2 (1991), pp. 31–46.MathSciNetGoogle Scholar
  10. [10]
    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.MathSciNetGoogle Scholar
  11. [11]
    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 (1993), pp.518–523.MathSciNetGoogle Scholar
  12. [12]
    Maslov, S. Yu. and Kurenkov, Yu. N.The increasing freedom of choice strategy for the satisfiability problem. In: “Proceedings of the USSR National Conference on Methods of Mathematical Logic in Artificial Intelligence and Programming”, Palanga, 1980 (in Russian).Google Scholar
  13. [13]
    Maslov, S. Yu.Theory of deductive systems and its applications. MIT Press, Cambridge, MA, 1987.Google Scholar
  14. [14]
    Moore, R. E.Methods and applications of interval analysis. SIAM, Philadelphia, 1979.Google Scholar
  15. [15]
    Poljak, S. and Rohn, J.Checking robust non-singularity is NP-hard. Mathematics of Control, Signals, and Systems6 (1993), pp. 1–9.CrossRefMathSciNetGoogle Scholar
  16. [16]
    Rall, L. B.Optimization of interval computations. In: Nickel, K. (ed.) “Interval Mathematics 1980” Academic Press, N.Y., 1980, pp. 489–498.Google Scholar
  17. [17]
    Ratschek, H. and Rokne, J.Optimality of the centered form. In: Nickel, K. (ed.) “Interval Mathematics 1980”, Academic Press, N.Y., 1980, pp. 499–508.Google Scholar
  18. [18]
    Rohn, J. and Kreinovich, V.Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard. SIAM Journal on Matrix Analysis and Applications (SIMAX)16(2) (1994), pp. 415–420.MathSciNetGoogle Scholar

Copyright information

© Institute of New Technologies in Education 1995

Authors and Affiliations

  • Bonnie Traylor
    • 1
  • Vladik Kreinovich
    • 2
  1. 1.M/S 301-270 Jet Propulsion LaboratoryPasadenaUSA
  2. 2.Computer Science DepartmentUniversity of Texas at El PasoEl PasoUSA

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