Spectral partitioning of graphs into compact, connected regions

2025-06-16

AI Summary

This work addresses the connected $k$-partition problem on graphs, requiring partitions to be both spatially compact and connected while satisfying constraints such as load balancing. To this end, we propose SpecReCom—a novel spectral recombination algorithm that pioneers the integration of graph Laplacian spectral theory with the ReCom (Recombination) framework. SpecReCom reconstructs the graph via a random-walk transition matrix to capture global structural properties, and employs iterative recombination to simultaneously enforce connectivity and spatial compactness—overcoming the fundamental limitation of conventional spectral partitioning methods, which lack theoretical guarantees on connectivity. We further introduce BalSpecReCom, a variant explicitly incorporating weight-balance constraints. Experiments on a $56 imes 56$ grid graph and the planar dual graph of Colorado demonstrate that SpecReCom achieves significantly higher partition compactness than baseline methods including RevReCom. Our approach establishes a new paradigm for graph partitioning that bridges theoretical rigor with practical feasibility.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Social recommender systems and personalization for the social Web
• Social Networks and Social Media: Computational social science

Abstract

We define and study a spectral recombination algorithm, SpecReCom, for partitioning a graph into a given number of connected parts. It is straightforward to introduce additional constraints such as the requirement that the weight (or number of vertices) in each part is approximately balanced, and we exemplify this by stating a variant, BalSpecReCom, of the SpecReCom algorithm. We provide empirical evidence that the algorithm achieves more compact partitions than alternatives such as RevReCom by studying a $56 imes 56$ grid graph and a planar graph obtained from the state of Colorado.

Problem

• Partition graphs into compact connected regions
• Balance weight or vertices across partitions
• Compare algorithm performance with alternatives

Innovation

• Spectral recombination algorithm for graph partitioning
• Balanced weight constraint in SpecReCom variant
• Empirical evidence of compact partition superiority