Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime
Abstract
The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity.
Cite
@article{arxiv.1211.4974,
title = {Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime},
author = {Akitoshi Kawamura and Norbert Th. Müller and Carsten Rösnick and Martin Ziegler},
journal= {arXiv preprint arXiv:1211.4974},
year = {2012}
}