First-principles multiway spectral partitioning of graphs

TL;DR

This paper introduces a first-principles spectral method for multiway graph partitioning, deriving the algorithm from a matrix formulation and relaxation, leading to improved performance on challenging problems.

Contribution

It presents a novel spectral partitioning algorithm derived from first principles, providing a clearer theoretical foundation and better results on difficult graph partitioning tasks.

Findings

Outperforms previous spectral methods on difficult partitioning problems

Provides a first-principles derivation of multiway spectral partitioning

Shows the effectiveness of the relaxed optimization approach

Abstract

We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. While there exist well-established spectral algorithms for this problem that give good results, they have traditionally not been well motivated. Rather than being derived from first principles by minimizing graph cuts, they are typically presented without direct derivation and then proved after the fact to work. In this paper, we take a contrasting approach in which we start with a matrix formulation of the minimum cut problem and then show, via a relaxed optimization, how it can be mapped onto a spectral embedding defined by the leading eigenvectors of the graph Laplacian. The end result is an algorithm that is similar in spirit to, but different in detail from, previous spectral partitioning approaches. In tests of the algorithm we find that it outperforms previous approaches on certain particularly difficult partitioning problems.