Highly Connected Graph Partitioning: Exact Formulation and Solution Methods

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.