Improvements on Spectral Bisection

TL;DR

This paper introduces a new spectral bisection algorithm utilizing both the second and third eigenvectors of the Laplacian matrix, improving cut quality over traditional methods and enabling partition refinement.

Contribution

It demonstrates the structural significance of the third eigenvector and develops a novel spectral bisection algorithm that guarantees smaller cuts than classic methods.

Findings

New spectral bisection algorithm using second and third eigenvectors

Algorithm guarantees smaller or equal cuts compared to classic spectral bisection

Provides a spectral refinement method for existing partitions

Abstract

We investigate combinatorial properties of certain configurations of a graph partition which are related to the minimality of a cut. We show that such configurations are related to the third eigenvector of the Laplacian matrix. It is well known that the second eigenvector encodes structural information, and that can be used to approximate a minimum bisection. In this paper, we show that the third eigenvector carries structural information as well. We then provide a new spectral bisection algorithm using both eigenvectors. The new algorithm is guaranteed to return a cut that is smaller or equal to the one returned by the classic spectral bisection. Also, we provide a spectral algorithm that can refine a given partition and produce a smaller cut.