Fundamental to many applications in data analysis are the decompositions of a graph, i.e. partitions of the node set into component-inducing subsets. One way of encoding decompositions is by multicuts, the subsets of those edges that straddle distinct components. Recently, a lifting of multicuts from a graph $G=(V,E)$ to an augmented graph $\widehat{G}=(V,E\cup F)$ has been proposed in the field of image analysis, with the goal of obtaining a more expressive characterization of graph decompositions in which it is made explicit also for pairs $F\subseteq \left(\genfrac{}{}{0.0pt}{}{V}{2}\right)\setminus E$ of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in ${\mathbb{R}}^{E\cup F}$ whose vertices are precisely the characteristic vectors of multicuts of $\widehat{G}$ lifted from $G$, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.

数据分析中许多应用的基础是图的分解，即将节点集划分为组件诱导子集。编码分解的一种方法是通过多切割，即跨越不同组件的那些边的子集。最近，从图中提升多割$G=(V,乙)$到增强图$\widehat{G}=(V,乙\cup F)$已在图像分析领域提出，目标是获得图形分解的更具表现力的特征，其中对对也明确$F\subseteq \left(\genfrac{}{}{0.0pt}{}{V}{\mathrm{2个}}\right)\setminus 乙$非相邻节点的数量，无论它们是在相同的还是不同的组件中。在这项工作中，我们详细研究了${\mathbb{R}}^{乙\cup F}$其顶点恰好是多割的特征向量$\widehat{G}$从$G$，特别是将其与先前关于集团划分和多线性多胞形的丰富工作联系起来。