A conjecture of Milena Mihail and Umesh Vazirani (Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.) states that the edge expansion of the graph of every $0/1$ polytope is at least one. Any lower bound on the edge expansion gives an upper bound for the mixing time of a random walk on the graph of the polytope. Such random walks are important because they can be used to generate an element from a set of combinatorial objects uniformly at random. A weaker form of the conjecture of Mihail and Vazirani says that the edge expansion of the graph of a $0/1$ polytope in ${\mathrm{ℝ}}^d$ is greater than one over some polynomial function of $d$. This weaker version of the conjecture would suffice for all applications. Our main result is that the edge expansion of the graph of a random $0/1$ polytope in ${\mathrm{ℝ}}^d$ is at least $\frac{1}{12d}$ with high probability.

中文翻译：

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随机 0/1 多胞形的扩展

Milena Mihail 和 Umesh Vazirani 的猜想（Proc. 24th Annu. ACM Symp. Theory Comput., ACM, Victoria, BC, 1992, pp. 26–38.）指出，每个图的边展开$0/1$多胞体至少是一个。边缘扩展的任何下限给出了多面体图上随机游走的混合时间的上限。这种随机游走很重要，因为它们可用于从一组组合对象中均匀随机地生成元素。Mihail 和 Vazirani 猜想的一个较弱形式表示，a 图的边展开$0/1$多胞体中${\mathrm{ℝ}}^d$大于某个多项式函数的一$d$。这个猜想的较弱版本足以满足所有应用。我们的主要结果是随机图的边扩展 $0/1$多胞体中${\mathrm{ℝ}}^d$至少是$\frac{1}{12d}$有很高的概率。