Multiway Spectral Graph Partitioning: Cut Functions, Cheeger Inequalities, and a Simple Algorithm

TL;DR

This paper presents a spectral graph partitioning method using eigenvalues and eigenvectors, establishing Cheeger inequalities and demonstrating an efficient, robust algorithm for multiway graph partitioning.

Contribution

It introduces a new spectral approach for multiway graph partitioning, linking eigenvalues to cut functions with Cheeger inequalities and a simple, effective algorithm.

Findings

The spectral method accurately identifies graph partitions.

Cheeger inequalities relate eigenvalues to partition quality.

Numerical examples show the algorithm's efficiency and robustness.

Abstract

The problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the k > 2 largest eigenvalues of the normalized adjacency matrix (equivalently, the k smallest eigenvalues of the normalized graph Laplacian) are computed. It is shown that the information necessary for partitioning is contained in the subspace spanned by the k eigenvectors. The partitioning is encoded in a matrix $\Psi$ in indicator form, which is computed by approximating the eigenvector matrix by a product of $\Psi$ and an orthogonal matrix. A measure of the distance of a graph to being k-partitionable is defined, as well as two cut (cost) functions, for which Cheeger inequalities are proved; thus the relation between the eigenvalue and partitioning problems is established. Numerical examples are given that demonstrate that the partitioning algorithm is efficient and robust.