Box Facets and Cut Facets of Lifted Multicut Polytopes

TL;DR

This work analyzes the lifted multicut polytope $\Xi_{G\widehat{G}}$ arising from a graph $G=(V,E)$ and its augmentation $\widehat{G}=(V,E\cup F)$. It provides a necessary, sufficient, and efficiently decidable condition for when a lower box inequality $0 \leq x_{uw}$ defines a facet of $\Xi_{G\widehat{G}}$, using polyhedral arguments and Menger’s theorem. It also proves that deciding facet-definingness for cut inequalities is NP-hard, via a reduction framework including a $f_d$-path construction and a 3-SAT reduction, highlighting intrinsic complexity in the cut-structure of lifted multicut polytopes. Collectively, the results sharpen understanding of the facet structure and have implications for designing effective cutting-plane algorithms for the lifted multicut problem. They also suggest future work on separator-based structures and non-local connectedness to further refine facet characterizations.

Abstract

The lifted multicut problem is a combinatorial optimization problem whose feasible solutions relate one-to-one 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 NP-hard.

Figures (5)

• Figure 1: Depicted on the left is a graph $G = (V, E)$ with $E = \{e_1, e_2\}$ and an augmentation $\widehat{G} = {(V, E \cup F)}$ of $G$ with $F = \{f\}$. Depicted in the middle are the four feasible solutions to the lifted multicut problem with respect to $G$ and $\widehat{G}$. Depicted on the right is the lifted multicut polytope $\Xi_{G \widehat{G}}$. The figure is adopted from andres-2023-a-polyhedral.
• Figure 2: Depicted above are two examples of a graph $G$ (solid edges) and augmentation $\widehat{G}$ (dashed edges) such that a condition of \ref{['theorem:box-inequalities']} is violated for the inequality $0 \leq x_{uw}$. In the upper example, the path with the edge set $\{uv_0, v_0v_1, v_1v_2, v_2v_3, v_3v_4, v_4w\}$ violates \ref{['theorem:condition:path']}. In the lower example, the cycle with the edge set $\{v_0v_1, v_1v_2, v_2v_3, v_3v_4, v_4v_5, v_5v_0\}$ violates \ref{['theorem:condition:cycle']}. For both cases, Equation \ref{['equation:equality']} from the proof of \ref{['theorem:box-inequalities']} is stated. Edges depicted in green occur with a positive sign in this equation, and edges depicted in red occur with a negative sign.
• Figure 3: Depicted on the left is a graph $G$, and depicted on the right is the corresponding auxiliary graph $G'$, whose construction is described in the proof of \ref{['lemma:cycle-even-length']}. The nodes depicted in green are proper $uw$-cut-nodes of $G$ and get removed in the construction of $G'$.
• Figure 4: Depicted above is an example of a graph $G$ (solid edges) and augmentation $\widehat{G}$ (dashed edges) that fulfills the conditions of \ref{['theorem:box-inequalities']} for $0 \leq x_{uw}$. Essential for the sufficiency proof of this theorem is that the introduced edge sets $H$ and $H_j$ for $j \in \mathbb{N}_0$ have the property $H \subseteq \cup_{j \geq 0} H_j$. In the given example, $H = {\{v_0v_1, v_1v_2, v_2v_3, v_3v_4, v_4v_5\}}$, $H_0 = {\{v_0v_1, v_4v_5\}}$, $H_1 = {\{v_1v_2, v_3v_4\}}$, $H_2 = {\{v_2v_3\}}$, and $H_j = \emptyset$ for $j \geq 3$. Thus, $H \subseteq \cup_{j \geq 0} H_j$.
• Figure 5: Depicted above is an example of the reduction from 3-sat used in the proof of \ref{['lemma-complexity-fd']}. Graphs $G$ and $\widehat{G}$ are constructed from the instance of the 3-sat problem given by $\neg \, x_1 \lor x_2 \lor x_3$. The additional edge $f$ as well as the edges in the $f$-cut $\delta$ are depicted in red. The $f_d$-path with respect $\delta$, given by the green edges and $d$, corresponds to the solution of the 3-sat problem instance: $\varphi(x_1) = \textsc{false}, \, \varphi(x_2) = \textsc{false} \ \text{and} \ \varphi(x_3) = \textsc{true}$.

Theorems & Definitions (18)

• Definition 1.1
• Lemma 3.1
• proof
• Theorem 4.1
• Lemma 4.2
• proof : Proof of \ref{['lemma:cycle-even-length']}
• proof : Proof of \ref{['theorem:box-inequalities']}
• Claim 4.3
• Definition 5.1
• Lemma 5.2