发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了任意多个广义交叉多胞形析取的凸包，给出垂直面的显式构造、面数上界及多项式时间分离算法，并证明成对不等式在低维下充分但一般不足。
AI 中文摘要
我们继续研究由$n$个二元指示变量管理的$\mathbb{R}^d$中$n+1$个多胞形的析取在$\mathbb{R}^{d+n}$中的凸包$D$。此前，当$d\in\{1,2\}$时，$D$已对任意多胞形进行了刻画；对于共享公共约束矩阵（在包括超矩形在内的技术条件下）的多胞形，以及两个广义交叉多胞形（$n=1$）的情况也已刻画。这里，当所有$n+1$个多胞形均为广义交叉多胞形时，我们给出对任意$n$和$d$的$D$的完整描述。每个垂直面由归一化支撑不等式描述，且当且仅当该不等式的等式图（即多胞形与坐标上的二分图）连通时，该不等式才是刻画面；所有垂直面均由可行生成树显式生成。我们刻画了此类面何时为先前已知的成对不等式，并证明当$d\leq3$时成对不等式足够，但一般情况下并非如此。我们在垂直面与$\Delta_n\times\Delta_{d-1}$的正则细分最大胞腔之间建立双射，由此得出$D$的面数的尖锐上界$n+1+2^d\binom{n+d-1}{n}$，该上界在一般数据下达到。我们还给出一个多项式时间分离算法，返回一个违反的面，该面在归一化支持不等式中是最大违反的。计算实验表明，非成对不等式的面填补了成对不等式留下的大部分空白。
英文摘要
We continue the study of the convex hull $D\subset\mathbb{R}^{d+n}$ of a disjunction of $n+1$ polytopes in $\mathbb{R}^d$, managed by $n$ binary indicator variables. Previously, $D$ had been characterized for arbitrary polytopes when $d\in\{1,2\}$, for polytopes sharing a common constraint matrix (under a technical condition, including hyper-rectangles), and for two generalized cross polytopes ($n=1$). Here we give a complete description of $D$ for arbitrary $n$ and $d$ when all $n+1$ polytopes are generalized cross polytopes. Every vertical facet is described by a normalized support inequality, and such an inequality is facet describing if and only if its equality graph, a bipartite graph on the polytopes and the coordinates, is connected; all vertical facets are generated explicitly from feasible spanning trees. We characterize when such a facet is one of the previously known pairwise inequalities, and we show that pairwise inequalities suffice when $d\leq3$, but not in general. We establish a bijection between the vertical facets and the maximal cells of a regular subdivision of $Δ_n\timesΔ_{d-1}$\,, which yields the sharp bound $n+1+2^d\binom{n+d-1}{n}$ on the number of facets of $D$, attained for generic data. We also give a polynomial-time separation algorithm that returns a violated facet, which is maximally violated among the normalized support inequalities. Computational experiments show that the facets that are not pairwise inequalities close a large part of the gap left by the pairwise inequalities.
