Applications of Interval Computations pp 81-89 | Cite as

# Linear Interval Equations: Computing Enclosures with Bounded Relative Overestimation is NP-Hard

Chapter

## Abstract

It is proved that if there exists a polynomial-time algorithm for enclosing solutions of linear interval equations with relative overestimation better than \(\frac{4}{{{n^2}}}\) (where *n* is the number of equations), then P=NP. The result holds for the symmetric case as well.

## Keywords

Symmetric Matrice Symmetric Case Rational Bound Interval Matrix Linear Interval System
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## References

- [1]G. Alefeld and G. Mayer, “On the symmetric and unsymmetric solution set of interval systems,” to appear in
*SIAM J. Matr. Anal. Appl*.Google Scholar - [2]G. Alefeld and J. Herzberger,
*Introduction to Interval Computations*, Academic Press, N. Y. 1983.zbMATHGoogle Scholar - [3]M. E. Garey and D. S. Johnson,
*Computers and Intractability: A Guide to the Theory of NP-Completeness*, Freeman, San Francisco 1979.zbMATHGoogle Scholar - [4]G. H. Golub and C. F. van Loan,
*Matrix Computations*, The Johns Hopkins University Press, Baltimore, 1983.zbMATHGoogle Scholar - [5]C. Jansson, “Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side,”
*Computing*, 1991, Vol. 46, pp. 265–274.MathSciNetzbMATHCrossRefGoogle Scholar - [6]W. Oettli and W. Prager, “Compatibility of Approximate Solution of Linear Equations with Given Error Bounds for Coefficients and Right-hand Sides”,
*Numer. Math*., 1964, Vol. 6, pp. 405–409.MathSciNetzbMATHCrossRefGoogle Scholar - [7]A. Neumaier,
*Interval Methods for Systems of Equations*, Cambridge University Press, Cambridge 1990.zbMATHGoogle Scholar - [8]J. Rohn, “Systems of linear interval equations,”
*Lin. Alg. Appls*., 1989, Vol. 126, pp. 39–78.MathSciNetzbMATHCrossRefGoogle Scholar - [9]J. Rohn, “Checking positive definiteness or stability of symmetric interval matrices is NP-hard,”
*Commentat. Math. Univ. Carolinae*, 1994, Vol. 35, pp. 795–797.MathSciNetzbMATHGoogle Scholar - [10]J. Rohn, “NP-hardness results for some linear and quadratic problems,” Technical Report No. 619, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague 1995, 11 pp.Google Scholar
- [11]J. Rohn,
*Linear interval equations: computing sufficiently accurate enclosures is NP-hard*, Technical Report No. 621, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague 1995, 7 pp.Google Scholar - [12]J. Rohn and V. Kreinovich, “Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard,”
*SIAM J. Matr. Anal Appl*, 1995, Vol. 16, pp. 415–420.MathSciNetzbMATHCrossRefGoogle Scholar

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© Kluwer Academic Publishers 1996