Eigenvalues, eigenvectors, and graph partitioning

Abstract

Spectral partitioning methods use the Fiedler vector—the eigenvector of the second-smallest eigenvalue of the Laplacian matrix—to find a small separator of a graph. These methods are important components of many circuit design and scientific numerical algorithms, and have been demonstrated by experiment to work extremely well. However, the quality of the partition that these methods should produce had eluded precise analysis.In this talk, we show that the proper application of spectral partitioning techniques works well on bounded-degree planar graphs and finite element meshes—the classes of graphs to which they are usually applied. In particular, we prove that the Fiedler vector can be used to produce separators whose ratio of vertices removed to edges cut is ø/trn for bounded-degree planar graphs and two-dimensional meshes and Øn 1/d for well-shaped d-dimensional meshes. The main ingredient of our analysis is a new geometric technique for estimating the second-smallest eigenvalues of the Laplacian matrices of these graphs.We will also explain why naive applications of spectral partitioning, such as spectral bisection, will fail miserably on some graphs that could conceivably arise in practice.