常余秩多面体上的随机游动
Random Walks on Polytopes of Constant Corank
摘要：
我们证明了$r$维空间中的一条直线和$n$点的旋转过程可能需要$Ω(\log^r n)$步骤作为$n \to \infty$。先前已知结合的唯一情况是$r \le 3$。我们的下界也是有效的预期数量的旋转步骤在以下应用：（1）随机边缘单纯形算法的线性规划与$n$约束的$d = n - r$变量;（2）有向随机游动的corank $r$网格多面体与$n$方面。
英文摘要：
We show that the pivoting process associated with one line and $n$ points in $r$-dimensional space may need $Ω(\log^r n)$ steps in expectation as $n \to \infty$. The only cases for which the bound was known previously were for $r \le 3$. Our lower bound is also valid for the expected number of pivoting steps in the following applications: (1) The Random-Edge simplex algorithm on linear programs with $n$ constraints in $d = n - r$ variables; and (2) the directed random walk on a grid polytope of corank $r$ with $n$ facets.
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