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

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Reliable Computing

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.

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  1. Department of Mathematics and Computer Science, Syddansk Universitet, Campusvej 55, 5230, Odense M, Denmark, e-mail

    Klaus Meer

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  1. Klaus Meer
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Meer, K. On a Refined Analysis of Some Problems in Interval Arithmetic Using Real Number Complexity Theory. Reliable Computing 10, 209–225 (2004). https://doi.org/10.1023/B:REOM.0000032109.55408.1a

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  • DOI: https://doi.org/10.1023/B:REOM.0000032109.55408.1a

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