Highly Connected Graph Partitioning: Exact Formulation and Solution Methods

TL;DR

This work defines Highly Connected Graph Partitioning (HCGP), which partitions a graph into $K$ $L,U$-balanced parts that are $Q$-connected and minimizes a compactness (k-median) objective. It develops an extensible IP formulation (the $Q$-CUT model) built on a Hess PDP foundation, augmented with a polynomial-time-separable separation routine and minimum-degree cuts, plus an exact branch-and-cut solver and a specialized ear-construction heuristic for $Q=2$. Key contributions include a general IP framework for any $Q\ge1$, a separator-based exact method with on-the-fly constraint generation, a dedicated heuristic for $Q=2$, and comprehensive computational results on 42 real-world graphs demonstrating the trade-offs between connectivity level, solution quality, and computation time. The approach enables practical, fault-tolerant graph partitions applicable to social, power, and transportation networks, with quantified costs for enforcing higher connectivity.

Abstract

Graph partitioning (GP) and vertex connectivity have traditionally been two distinct fields of study. This paper introduces the highly connected graph partitioning (HCGP) problem, which partitions a graph into compact, size balanced, and $Q$-(vertex) connected parts for any $Q\geq 1$. This problem is valuable in applications that seek cohesion and fault-tolerance within their parts, such as community detection in social networks and resiliency-focused partitioning of power networks. Existing research in this fundamental interconnection primarily focuses on providing theoretical existence guarantees of highly connected partitions for a limited set of dense graphs, and do not include canonical GP considerations such as size balance and compactness. This paper's key contribution is providing a general modeling and algorithmic approach for HCGP, inspired by recent work in the political districting problem, a special case of HCGP with $Q=1$. This approach models $Q$-connectivity constraints as mixed integer programs for any $Q\geq 1$ and provides an efficient branch-and-cut method to solve HCGP. When solution time is a priority over optimality, this paper provides a heuristic method specifically designed for HCGP with $Q=2$. A computational analysis evaluates these methods using a test bed of instances from various real-world graphs. In this analysis, the branch-and-cut method finds an optimal solution within one hour in $82.8\%$ of the instances solved. For $Q=2$, small and sparse instances are challenging for the heuristic, whereas large and sparse instances are challenging for the exact method. Furthermore, this study quantifies the computational cost of ensuring higher connectivity using the branch-and-cut approach, compared to a baseline of ensuring $1$-connectivity. Overall, this work serves as an effective tool to partition a graph into resilient and cohesive parts.

Figures (12)

• Figure 1: Graph examples illustrating vertex connectivity; shaded vertices constitute a minimum cutset.
• Figure 2: $1$-connected versus $3$-connected parts: An example illustrating optimal $5,5$-balanced $Q$-proper $2$-partitions of a graph $G$ for $Q \in \{1,3\}$; assume each vertex's weight to be one. Solid lines are internal edges in each part and dotted lines are inter-part edges. Each part's root vertex is shaded.
• Figure 3: A minimal $a,b$-separator represented by shaded vertices in a $5\times 5$ grid graph.
• Figure 4: Abstract representations of a $Q$-disconnected part with root $r$ in graph $G$ considering three cases corresponding to the three sets of constraints added in Algorithm \ref{['algo_intseparation_biconnected']}.
• Figure 5: An open ear decomposition of a $2$-connected graph with $10$ vertices into four ears.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Theorem 1
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• Theorem 3
• Definition 3
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