How to Integrate a Polynomial over a Simplex

TL;DR

This paper analyzes the computational complexity of integrating polynomials over simplexes, establishing NP-hardness in general but providing efficient algorithms for fixed-variable cases, with extensions and experimental insights.

Contribution

It proves NP-hardness for general polynomial integration over simplexes and offers polynomial-time algorithms when the polynomial depends on a fixed number of variables.

Findings

NP-hardness for arbitrary polynomials

Polynomial-time algorithms for fixed-variable polynomials

Extensions to other polytopes and experimental results

Abstract

This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes, discussion of other available methods and experimental results.