Range Estimation is NP-Hard for Epsilon-Square Accuracy and Feasible for Epsilon to the Power of 2-Delta

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

4-2002

Comments

UTEP-CS-99-45a.

Published in Reliable Computing, 2002, Vol. 8, No. 6, pp. 481-491.

Abstract

The basic problem of interval computations is: given a function f(x1,...,xn) and n intervals [xi-,xi+], find the (interval) range Y of the given function on the given intervals. It is known that even for quadratic polynomials f(x1,...,xn), 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 "epsilon" 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 times epsilon squared, then the problem is still NP-hard; on the other hand, for every delta>0, there exists a feasible algorithm which asymptotically, estimates the range with an accuracy c times epsilon to the power 2-delta.