论文标题
晶格多面体的小阴影
Small Shadows of Lattice Polytopes
论文作者
论文摘要
$ d $维晶格$ p \ subseteq [0，k]^{n} $的图表的直径最多是$ dk $，这是由于Kleinschmidt和onn的工作。但是，这是一个空旷的问题，单调直径是单调路径的最短长度，是$ d $ - 二维的晶格polytope $ p = \ {\ mathbf {x}：a \ mathbf {x} \ leq \ leq \ leq \ leq \ mathbf {b} $ d $和$ k $中的多项式。这个问题在线性优化中特别感兴趣，因为单纯形方法所追踪的路径必须单调。我们在这个方向上介绍部分结果，包括$ k = 2 $ $ 3D $的单调直径，单调直径为$（d-1）m+1 $ for $ d $ d $ - 二维$（m+1）$ - 级别poltytopes polytopes，pivot polytopes，pivot pivot规则，是一种pivot规则，可以保证$ dnk $ dnk | n ynk y ynydemerive y ynydevery y ryy dry n | y | $ p $的LP，以及从某些固定起点的路径长度的$ dk $限制。最后，我们提出了一种建设性的方法，以$（3/2）dk $的直径结合，并描述如何将此最终绑定到通过追踪这样的路径来解决线性程序的算法。
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.
