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.
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Kreinovich, V. Range Estimation Is NP-Hard for ε2 Accuracy and Feasible for ε2−δ . Reliable Computing 8, 481–491 (2002). https://doi.org/10.1023/A:1021368627321
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DOI: https://doi.org/10.1023/A:1021368627321