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有广泛应用。基于线性规划的精确算法需要理解提升多割多面体。尽管近期取得进展，关于这些多面体的两个基本问题仍未解决：哪些下立方体不等式定义多面体面，以及哪些切不等式定义多面体面？在本文中，我们通过建立必要、充分且可高效判定的条件回答了第一个问题。针对第二个问题，我们证明了判定切不等式是否定义多面体面是NP难的。这完成了对提升多割多面体典型多面体面的完整分析。