We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for integer polytopes in terms of the length of the description of the polytope (in bits) and the minimum angle between facets of its polar. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure $1-o(1)$ and polynomial diameter. Both bounds rely on spectral gaps -- of a certain Schr\"odinger operator in the first case, and a certain continuous time Markov chain in the second -- which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.
翻译:我们用两个设置来证明多面体的图形直径的上界。 第一个是按多极点( 以位数计) 描述长度和极间最小角度的整数多面体最差的框框。 第二个是平滑的捆绑分析 : 给一个适当正常化的多面体, 我们给每个约束度添加小高斯语噪音。 我们考虑在环绕多极点体的顶部( 对应其极点的平均曲线度量) 上进行自然几何测量, 并显示极点有极点的“ 引力元件 ”, 高度可能存在1美元-1美元和多面直径的“ 引力元件 ” 。 两种捆绑都依赖于光谱的缺口 -- 第一个是某个Schr\“ odiginger” 操作员, 第二个是一定的连续时间 Markov 链 -- 这是简单的多极点体体体体量的逻辑一致性, 其变软变量产生的 。