The lifted multicut problem is a combinatorial optimization problem whose feasible solutions relate one-to-one to the decompositions of a graph $G = (V, E)$. Given an augmentation $\widehat{G} = (V, E \cup F)$ of $G$ and given costs $c \in \mathbb{R}^{E \cup F}$, the objective is to minimize the sum of those $c_{uw}$ with $uw \in E \cup F$ for which $u$ and $w$ are in distinct components. For $F = \emptyset$, the problem specializes to the multicut problem, and for $E = \tbinom{V}{2}$ to the clique partitioning problem. We study a binary linear program formulation of the lifted multicut problem. More specifically, we contribute to the analysis of the associated lifted multicut polytopes: Firstly, we establish a necessary, sufficient and efficiently decidable condition for a lower box inequality to define a facet. Secondly, we show that deciding whether a cut inequality of the binary linear program defines a facet is NP-hard.

翻译：提升多割问题是一个组合优化问题，其可行解与图 $G = (V, E)$ 的分解一一对应。给定 $G$ 的增广图 $\widehat{G} = (V, E \cup F)$ 和成本 $c \in \mathbb{R}^{E \cup F}$，目标是最小化使得 $u$ 与 $w$ 位于不同连通分量的那些 $c_{uw}$（其中 $uw \in E \cup F$）之和。当 $F = \emptyset$ 时，该问题特化为多割问题；当 $E = \tbinom{V}{2}$ 时，则特化为团划分问题。我们研究了提升多割问题的二元线性规划形式。具体而言，我们对关联的提升多割多面体的分析作出以下贡献：首先，我们建立了下盒不等式定义面的一个必要、充分且可高效判定的条件。其次，我们证明判定二元线性规划的切割不等式是否定义面是NP难的。