Computing the partition function of a polynomial on the Boolean cube

TL;DR

This paper introduces a quasi-polynomial time algorithm to approximate the partition function of polynomials on the Boolean cube, with applications to maximizing such polynomials under certain conditions.

Contribution

It presents the first quasi-polynomial algorithm for approximating the partition function of polynomials on the Boolean cube, extending to maximum value approximation under specific constraints.

Findings

Algorithm approximates the partition function within relative error epsilon.

Applicable when |lambda| < 1/(2 L sqrt{deg f}) and L bounds the Lipschitz constant.

Provides approximation guarantees for polynomial maximization problems with bounded variable participation.

Abstract

For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda and epsilon >0 approximates the partition function within a relative error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f from above and N is the number of monomials in f. As a corollary, we obtain a quasi-polynomial algorithm, which, given such an f with coefficients +1 and -1 and such that every variable enters not more than 4 monomials, approximates the maximum of f on {-1, 1}^n within a factor of O(sqrt{deg f}/delta), provided the maximum is N delta for some 0< delta <1. If every variable enters not more than k monomials for some fixed k > 4, we are able to establish a similar result when delta > (k-1)/k.