Box Facets and Cut Facets of Lifted Multicut Polytopes

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 box 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.