格多面体的小阴影
Small Shadows of Lattice Polytopes
摘要：
由于Kleinschlaun和Onn的工作，已知$d$维晶格多面体$P \subseteq [0,k]^{n}$的图的直径至多为$dk$。然而，在$d$和$k$中，$d$维格多面体$P = \{\mathbf{x}: A\mathbf{x} \leq \mathbf{b}\} \subseteq [0,k]^{n}$的单调直径（单调路径的最短保证长度）是否受多项式约束是一个悬而未决的问题.这个问题在线性最优化中特别有趣，因为单纯形方法跟踪的路径必须是单调的。在这个方向上，我们介绍了部分结果，包括一个单调的$3d$的$k = 2$，$(d-1)m+1$的$d$ -维$(m+1)$ -水平多面体的单调直径界，枢轴规则，这样的单纯形方法是保证采取最多$dnk||A||_{\infty}$非退化的步骤，以解决LP $P$，和一个界的$dk$的路径长度从某些固定的起点。最后，我们提出了一个建设性的方法，直径界的$(3/2)dk$，并描述了如何将这个最终的界限转化为一个算法，解决了线性规划跟踪这样的路径。
英文摘要：
The diameter of the graph of a $d$-dimensional lattice polytope $P \subseteq [0,k]^{n}$ is known to be at most $dk$ due to work by Kleinschmidt and Onn. However, it is an open question whether the monotone diameter, the shortest guaranteed length of a monotone path, of a $d$-dimensional lattice polytope $P = \{\mathbf{x}: A\mathbf{x} \leq \mathbf{b}\} \subseteq [0,k]^{n}$ is bounded by a polynomial in $d$ and $k$. This question is of particular interest in linear optimization, since paths traced by the Simplex method must be monotone. We introduce partial results in this direction including a monotone diameter bound of $3d$ for $k = 2$, a monotone diameter bound of $(d-1)m+1$ for $d$-dimensional $(m+1)$-level polytopes, a pivot rule such that the Simplex method is guaranteed to take at most $dnk||A||_{\infty}$ non-degenerate steps to solve a LP on $P$, and a bound of $dk$ for lengths of paths from certain fixed starting points. Finally, we present a constructive approach to a diameter bound of $(3/2)dk$ and describe how to translate this final bound into an algorithm that solves a linear program by tracing such a path.
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