A polyhedral study of lifted multicuts

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 $\hat{G}=(V,E\cup F)$ 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 $F\subseteq \left(\frac{V}{2}\right)\setminus E$ of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in $R^{E\cup F}$ whose vertices are precisely the characteristic vectors of multicuts of $\hat{G}$ lifted from $G$, connecting it, in particular, to the rich body of prior work on the clique partitioning and multilinear polytope.