A Polyhedral Study of Lifted Multicuts

TL;DR

This paper investigates the polyhedral structure of lifted multicuts in augmented graphs, enhancing the understanding of graph decompositions with explicit non-neighboring node relationships.

Contribution

It provides a detailed polyhedral analysis of lifted multicuts, linking them to clique partitioning and multilinear polytopes, and advances the theoretical framework for graph decompositions.

Findings

Characterization of the polytope of lifted multicuts.

Connection to clique partitioning and multilinear polytopes.

Insights into the structure of graph decompositions.

Abstract

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.