On Approximating Univariate NP-Hard Integrals
For theoretical computer science and numerical analysis, it establishes fundamental hardness results for a class of integrals, showing that efficient numerical methods cannot achieve the convergence rates claimed in the literature without solving major complexity problems.
The paper proves that approximating certain univariate integrals involving products of cosines is #P-hard, and deciding whether such integrals are zero or infinity is NP-complete. This implies that claimed exponential convergence rates for numerical integration methods would imply P=#P, an impossibility.
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.