Optimizing Connected Components Graph Partitioning With Minimum Size Constraints Using Integer Programming and Spectral Clustering Techniques

TL;DR

This paper introduces a novel approach combining integer programming and spectral clustering to optimize graph partitioning into connected components with size constraints, applicable in various real-world scenarios.

Contribution

It presents a new mixed integer programming formulation with valid inequalities and a spectral clustering-assisted column generation method for constrained graph partitioning.

Findings

Column generation with spectral clustering improves solution efficiency.

Proposed methods effectively handle large and complex graph instances.

Computational results demonstrate the approach's potential in practical applications.

Abstract

In this work, a graph partitioning problem in a fixed number of connected components is considered. Given an undirected graph with costs on the edges, the problem consists of partitioning the set of nodes into a fixed number of subsets with minimum size, where each subset induces a connected subgraph with minimal edge cost. The problem naturally surges in applications where connectivity is essential, such as cluster detection in social networks, political districting, sports team realignment, and energy distribution. Mixed Integer Programming formulations together with a variety of valid inequalities are demonstrated and computationally tested. An assisted column generation approach by spectral clustering is also proposed for this problem with additional valid inequalities. Finally, the methods are tested for several simulated instances, and computational results are discussed. Overall, the proposed column generation technique enhanced by spectral clustering offers a promising approach to solve clustering and partitioning problems.