Partial Optimality in Cubic Correlation Clustering

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

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 $F \subseteq \tbinom{V}{2} \setminus E$ of non-neighboring nodes whether these are in the same or distinct components. In this work, we study in detail the polytope in $\mathbb{R}^{E \cup F}$ 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.