# Computing the Partition Function of a Polynomial on the Boolean Cube

• Alexander Barvinok
Chapter

## 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 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.

## 1991 Mathematics Subject Classification.

90C09 68C25 68W25 68R05

## Notes

### Acknowledgements

I am grateful to Johan Håstad for advice and references on optimizing a polynomial on the Boolean cube and to the anonymous referees for careful reading of the paper and useful suggestions.

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