Work

How to integrate a polynomial over a simplex

Year: 2010

Type: article

Abstract: This paper starts by settling the computational complexity of the problem of integrating a polynomial function f f over a rational simplex. We prove that the problem is N P \mathrm {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 explore our algorithms with some experiments. We conclude the article with extensions to other polytopes and discussion of other available methods.

Source: Mathematics of Computation

DOI: 10.1090/s0025-5718-2010-02378-6

Authors: Velleda Baldoni, Nicole Berline, Jesus De Loera, Matthias Köppe, Michèle Vergne

Institutions: University of Rome Tor Vergata, École Polytechnique, Centre de Mathématiques Laurent Schwartz, University of California, Davis, Institut de Mathématiques de Jussieu-Paris Rive Gauche

Language: English

Topic: Complexity and Algorithms in Graphs

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: bronze

Funders: National Science Foundation, Scuola Normale Superiore

Award: Algebraic Algorithms in Discrete Optimization and Tools for Computational Convexity