Box Facets and Cut Facets of Lifted Multicut Polytopes
Abstract
The lifted multicut problem is a combinatorial optimization problem whose feasible solutions relate onetoone to the decompositions of a graph $G = (V, E)$. Given an augmentation $\widehat{G} = (V, E \cup F)$ of $G$ and given costs $c \in \mathbb{R}^{E \cup F}$, the objective is to minimize the sum of those $c_{uw}$ with $uw \in E \cup F$ for which $u$ and $w$ are in distinct components. For $F = \emptyset$, the problem specializes to the multicut problem, and for $E = \tbinom{V}{2}$ to the clique partitioning problem. We study a binary linear program formulation of the lifted multicut problem. More specifically, we contribute to the analysis of the associated lifted multicut polytopes: Firstly, we establish a necessary, sufficient and efficiently decidable condition for a lower box inequality to define a facet. Secondly, we show that deciding whether a cut inequality of the binary linear program defines a facet is NPhard.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.16814
 arXiv:
 arXiv:2402.16814
 Bibcode:
 2024arXiv240216814F
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Machine Learning
 EPrint:
 10 pages, 5 figures