Computing the Partition Function of a Polynomial on the Boolean Cube

Abstract

For a polynomial $f:\{ -1,1\}^{n}\longrightarrow \mathbb{C}$, we define the partition function as the average of e ^λf(x) over all points x ∈ {−1, 1}ⁿ, where $\lambda \in \mathbb{C}$ is a parameter. We present a quasi-polynomial algorithm, which, given such f, λ and ε > 0 approximates the partition function within a relative error of ε in N ^{O(lnn−lnε)} time provided $\vert \lambda \vert \leq (2L\sqrt{\deg f})^{-1}$, 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 such that every variable enters not more than 4 monomials, approximates the maximum of f on { − 1, 1}ⁿ within a factor of $O\left (\delta ^{-1}\sqrt{\deg f}\right )$, provided the maximum is Nδ for some 0 < δ ≤ 1. If every variable enters not more than k monomials for some fixed k > 4, we are able to establish a similar result when δ ≥ (k − 1)∕k.

This research was partially supported by NSF Grant DMS 1361541.