多项式赫希猜想的任何证明都必然是完全不连贯的
Any Proof of Polynomial Hirsch Must be Completely Incoherent
浏览论文内容
中文总结 AI 辅助
研究多项式赫希猜想类似问题，即多面体上由线性函数诱导定向时是否总有多项式长度连贯单调路径。通过展示特定多面体和线性函数，证明并非如此，且加强了相关领域长期结果。
中文摘要 AI 辅助
1992年，比勒拉和施图姆费尔斯在描述纤维多面体构造时引入了多面体上的连贯单调路径，1994年与卡普拉诺夫证明这些连贯单调路径捕捉了由线性函数诱导定向的多面体有向图中所有单调路径（从最小值到最大值的路径）空间的拓扑结构。这些结果引出了多项式赫希猜想的类似问题：对于由线性函数诱导的任何定向选择，多面体上是否总是存在多项式长度的连贯单调路径？我们通过展示一族多面体和相应线性函数表明并非如此，其中每个连贯单调路径都是指数长的。作为应用，我们加强了关于影子单纯形法下界、离散几何中的几何横截以及参数线性优化的长期结果。
英文摘要
In 1992, Billera and Sturmfels introduced coherent monotone paths on polytopes as part of their description of the fiber polytope construction, and later in 1994 showed with Kapranov that these coherent monotone paths capture the topology of the space of all monotone paths, paths from a minimum to a maximum, in the directed graph of a polytope with orientation induced by a linear function. Those results motivate the following analog of the polynomial Hirsch conjecture: Does there always exist a coherent monotone path of polynomial length on a polytope for any choice of orientation induced by a linear function? We show this is not the case by exhibiting a family of polytopes and corresponding linear functions for which every coherent monotone path is exponentially long. As applications, we strengthen longstanding results pertaining to lower bounds for the shadow simplex method, geometric transversals in discrete geometry, and parametric linear optimization.
