On the connected (sub)partition polytope

TL;DR

The paper studies the connected $k$-subpartition polytope $\mathcal{P}(G,k)$, extending polyhedral insights from the $k=1$ case to multi-class partitions. It introduces single-class and multiclass valid inequalities, establishes facet-defining conditions (including lifting from $\mathcal{P}(G,1)$ and specialized tree results), and develops separation algorithms with practical complexity guarantees. It also proves NP-hardness for certain multiclass separations while providing polynomial-time solutions on trees for specific inequalities, and validates the approaches via computational experiments that show notable improvements in solving time and gap reduction. Together, these results advance both theoretical understanding and practical MILP formulations for connected partition problems with broad applications.

Abstract

Let $k$ be a positive integer and let $G$ be a graph with $n$ vertices. A connected $k$-subpartition of $G$ is a collection of $k$ pairwise disjoint sets (a.k.a. classes) of vertices in $G$ such that each set induces a connected subgraph. The connected $k$-subpartition polytope of $G$, denoted by $\poly(G,k)$, is defined as the convex hull of the incidence vectors of all connected $k$-subpartitions of $G$. Many applications arising in off-shore oil-drilling, forest planning, image processing, cluster analysis, political districting, police patrolling, and biology are modeled in terms of finding connected (sub)partitions of a graph. This study focuses on the facial structure of~$\poly(G,k)$ and the computational complexity of the corresponding separation problems. We first propose a set of valid inequalities having non-zero coefficients associated with a single class that extends and generalizes the ones in the literature of related problems, show sufficient conditions for these inequalities to be facet-defining, and design a polynomial-time separation algorithm for them. We also devise two sets of inequalities that consider multiple classes, prove when they define facets, and study the computational complexity of associated separation problems. Finally, we report on computational experiments showing the usefulness of the proposed inequalities.

Figures (12)

• Figure 1: Example of a fractional solution that violates a general connectivity inequality. (a) shows a graph $G$ with vertex weights given by a fractional solution in $\mathcal{P}'(G,k)$ for a fixed $c \in [k]$. (b) depicts an orientation $\vec{E}$ of $E(\mathcal{W})$, where the dotted rectangles represent the partition $\mathcal{W}=\{\{v_1,v_2\}, \{v_3\}, \{v_4,v_5\}\}$ and thicker circles the vertices in $S=\{v_1,v_3,v_5\}$.
• Figure 2: Example of a fractional solution in $\mathcal{P}'(G,k)$ that satisfies all indegree inequalities, but violates a general connectivity inequality. (a) shows a graph $G$ with vertex weights given by a fractional solution in $\mathcal{P}'(G,k) \cap \{x \in \mathbb{R}^{n}_\geq : x \text{ satisfies } \ref{['ineq:indegree']} \}$ for a fixed $c \in [k]$. (b) depicts an orientation $\vec{E}$ of $E(G)$ that maximizes the left-hand side of the indegree inequalities for that fractional solution. The dotted squares represent the partition $\mathcal{W}=\{\{v_1,v_2,v_3\}, \{v_4\}, \{v_5,v_6,v_7\}\}$, and thicker circles represent the vertices in $S= \{v_1,v_4,v_7\}$.
• Figure 3: Example illustrating the hangers in a directed graph.
• Figure 4: Examples of trees $T_v$ formed by hangers depicted in Figure \ref{['fig:example:hangers']}. Thicker circles identify the vertices in $S=\{v_1, v_4, v_7\}$.
• Figure 5: Examples of inequalities \ref{['ineq:multicolor-cut']}. Thicker vertices identifies the stable set, dotted rectangles represent the corresponding multiway cuts in the graphs and $C=\{a,b\}$.

Theorems & Definitions (34)

• Theorem 1: Campêlo et al.CAMPELO2013CAMPELO2016
• Lemma 1
• proof
• Corollary 1
• proof
• Theorem 2: Korte et al. korte2012greedoids (see wang2017imposing)
• Theorem 3
• proof
• Proposition 1
• proof