On Approximating Univariate NP-Hard Integrals

TL;DR

This paper explores the computational complexity of approximating certain univariate integrals, revealing their equivalence to #P and NP-Complete problems, and questions the accuracy of existing numerical methods due to these hardness results.

Contribution

The paper establishes the complexity-theoretic hardness of approximating specific integrals and challenges the validity of known convergence rates of classical numerical methods.

Findings

Approximating certain integrals is #P-hard.

Deciding if the integral is zero or infinite is NP-Complete.

Existing numerical methods' convergence rates are likely incorrect.

Abstract

Approximating a definite integral of product of cosines to within an accuracy of n binary digits where the integrand depends on input integers x[k] given in binary radix, is equivalent to counting the number of equal-sum partitions of the integers and is thus a #P problem. Similarly, integrating this function from zero to infinity and deciding whether the result is either zero or infinity is an NP-Complete problem. Efficient numerical integration methods such as the double exponential formula and the sinc approximation have been around since the mid 70's. Noting the hardness of approximating the integral we argue that the proven rates of convergence of such methods cannot possibly be correct since they give rise to an anomalous result as P=#P.