On the Optimality of the Median Cut Spectral Bisection Graph Partitioning Method

Authors:
Tony F. Chan;P. Ciarlet, Jr.;W. K. Szeto
Affiliations:
-;-;-
Venue:
SIAM Journal on Scientific Computing
Year:
1997

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Abstract

Recursive spectral bisection (RSB) is a heuristic technique for finding a minimum cut graph bisection. To use this method the second eigenvector of the Laplacian of the graph is computed and from it a bisection is obtained. The most common method is to use the median of the components of the second eigenvector to induce a bisection. We prove here that this median cut method is optimal in the sense that the partition vector induced by it is the closest partition vector, in any ls norm, for $s\ge1$, to the second eigenvector. Moreover, we prove that the same result also holds for any m-partition, that is, a partition into m and (n-m)$ vertices, when using the mth largest or smallest components of the second eigenvector.