The lifted multicut problem has diverse applications in the field of computer vision. Exact algorithms based on linear programming require an understanding of lifted multicut polytopes. Despite recent progress, two fundamental questions about these polytopes have remained open: Which lower cube inequalities define facets, and which cut inequalities define facets? In this article, we answer the first question by establishing conditions that are necessary, sufficient and efficiently decidable. Toward the second question, we show that deciding facet-definingness of cut inequalities is NP-hard. This completes the analysis of canonical facets of lifted multicut polytopes.

翻译：提升多割问题在计算机视觉领域具有多种应用。基于线性规划的精确算法要求理解提升多割多面体的结构。尽管近期取得了进展，关于这些多面体仍有两个基本问题未解决：哪些低维立方体不等式定义面，以及哪些割不等式定义面？在本文中，我们通过建立必要条件、充分条件及高效可判定条件，回答了第一个问题。针对第二个问题，我们证明判断割不等式是否定义面是NP困难的。这完成了对提升多割多面体典型面的分析。