# On a Refined Analysis of Some Problems in Interval Arithmetic Using Real Number Complexity Theory

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## Abstract

We study some problems in interval arithmetic treated in *Kreinovich et al.* [13]. First, we consider the best linear approximation of a quadratic interval function. Whereas this problem (as decision problem) is known to be *NP*-hard in the Turing model, we analyze its complexity in the real number model and the analogous class *NP*_{ℝ}. Our results substantiate that most likely it does not any longer capture the difficulty of *NP*_{ℝ} in such a real number setting. More precisely, we give upper complexity bounds for the approximation problem for interval functions by locating it in (a real analogue of). This result allows several conclusions:

• the problem is not (any more) *NP*_{ℝ}-hard under so called weak polynomial time reductions and likely not to be *NP*_{ℝ}-hard under (full) polynomial time reductions;

• for fixed dimension the problem is polynomial time solvable; this extends the results in *Koshelev et al.* [12] and answers a question left open in [13].

We also study several versions of interval linear systems and show similar results as for the approximation problem.

Our methods combine structural complexity theory with issues from semi-infinite optimization theory.

## Keywords

Approximation Problem Interval Function Complexity Theory Analogous Class Interval Arithmetic## Preview

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