Improvement of the Cascadic Multigrid Algorithm with a Gauss Seidel Smoother to Efficiently Compute the Fiedler Vector of a Graph Laplacian

Abstract

In this paper, we detail the improvement of the Cascadic Multigrid algorithm with the addition of the Gauss Seidel algorithm in order to compute the Fiedler vector of a graph Laplacian, which is the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in graph partitioning, particularly in the spectral clustering algorithm. The algorithm is algebraic and employs heavy edge coarsening, which was developed for the first cascadic multigrid algorithm. We present numerical tests that test the algorithm against a variety of matrices of different size and properties. We then test the algorithm on a range of square matrices with uniform properties in order to prove the linear complexity of the algorithm.