# Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ

Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ The basic problem of interval computations is: given a function f(x 1,..., x n) and n intervals [x i, x i], find the (interval) range yof the given function on the given intervals. It is known that even for quadratic polynomials f(x 1,..., x n), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound ε on the width of the input intervals tends to 0. We show that for small c > 0, if we want to compute the range with an accuracy c ⋅ ε2, then the problem is still NP-hard; on the other hand, for every δ > 0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c ⋅ ε2−δ. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ

Reliable Computing, Volume 8 (6) – Oct 13, 2004
11 pages

/lp/springer_journal/range-estimation-is-np-hard-for-2-accuracy-and-feasible-for-2-MVjiEVicRQ
Publisher
Springer Journals
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1021368627321
Publisher site
See Article on Publisher Site

### Abstract

The basic problem of interval computations is: given a function f(x 1,..., x n) and n intervals [x i, x i], find the (interval) range yof the given function on the given intervals. It is known that even for quadratic polynomials f(x 1,..., x n), this problem is NP-hard. In this paper, following the advice of A. Neumaier, we analyze the complexity of asymptotic range estimation, when the bound ε on the width of the input intervals tends to 0. We show that for small c > 0, if we want to compute the range with an accuracy c ⋅ ε2, then the problem is still NP-hard; on the other hand, for every δ > 0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c ⋅ ε2−δ.

### Journal

Reliable ComputingSpringer Journals

Published: Oct 13, 2004

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