A Bernoulli factory is an algorithmic procedure for exact sampling of certain random variables having only Bernoulli access to their parameters. Bernoulli access to a parameter $p \in [0,1]$ means the algorithm does not know $p$, but has sample access to independent draws of a Bernoulli random variable with mean equal to $p$. In this paper, we study the problem of Bernoulli factories for polytopes: given Bernoulli access to a vector $x\in \mathcal{P}$ for a given polytope $\mathcal{P}\subset [0,1]^n$, output a randomized vertex such that the expected value of the $i$-th coordinate is \emph{exactly} equal to $x_i$. For example, for the special case of the perfect matching polytope, one is given Bernoulli access to the entries of a doubly stochastic matrix $[x_{ij}]$ and asked to sample a matching such that the probability of each edge $(i,j)$ be present in the matching is exactly equal to $x_{ij}$. We show that a polytope $\mathcal{P}$ admits a Bernoulli factory if and and only if $\mathcal{P}$ is the intersection of $[0,1]^n$ with an affine subspace. Our construction is based on an algebraic formulation of the problem, involving identifying a family of Bernstein polynomials (one per vertex) that satisfy a certain algebraic identity on $\mathcal{P}$. The main technical tool behind our construction is a connection between these polynomials and the geometry of zonotope tilings. We apply these results to construct an explicit factory for the perfect matching polytope. The resulting factory is deeply connected to the combinatorial enumeration of arborescences and may be of independent interest. For the $k$-uniform matroid polytope, we recover a sampling procedure known in statistics as Sampford sampling.

中文翻译：

---

组合伯努利工厂：匹配、流动和其他多胞体

伯努利工厂是一种算法程序，用于对某些随机变量进行精确采样，只有伯努利才能访问其参数。伯努利对参数 $p \in [0,1]$ 的访问意味着该算法不知道 $p$，但可以对均值等于 $p$ 的伯努利随机变量的独立抽取进行样本访问。在本文中，我们研究了多胞体的伯努利工厂问题：给定伯努利访问向量 $x\in \mathcal{P}$ 对于给定的多胞体 $\mathcal{P}\subset [0,1]^n$ ，输出一个随机顶点，使得第 $i$ 个坐标的期望值 \emph{exactly} 等于 $x_i$。例如，对于完美匹配多胞体的特殊情况，给予伯努利访问双随机矩阵 $[x_{ij}]$ 的条目，并要求对匹配进行采样，使得每条边的概率 $(i , j)$ 出现在匹配中正好等于 $x_{ij}$。我们证明多面体 $\mathcal{P}$ 承认伯努利工厂当且仅当 $\mathcal{P}$ 是 $[0,1]^n$ 与仿射子空间的交集。我们的构造基于问题的代数公式，包括确定一系列满足 $\mathcal{P}$ 上特定代数恒等式的 Bernstein 多项式（每个顶点一个）。我们构建背后的主要技术工具是将这些多项式与带状切片的几何形状联系起来。我们应用这些结果来为完美匹配的多胞体构建一个明确的工厂。由此产生的工厂与树状的组合枚举密切相关，并且可能具有独立的兴趣。对于 $k$-uniform matroid polytope，