Spectral Partitioning: The More Eigenvectors, The Better

A spectral partitioning method uses the eigenvectors of a graph's adjacency or Laplacian matrix to construct a geometric representation (e.g., a linear ordering) which is then heuristically partitioned. We map each graph vertex to a vector in d-dimensional space, where d is the number of eigenvectors, such that these vectors constitute an instance of the vector partitioning problem. When all the eigenvectors are used, graph partitioning exactly reduces to vector partitioning. This result motivates a simple ordering heuristic that can be used to yield high-quality 2-way and multi-way partitionings. Our experiments suggest the vector partitioning perspective opens the door to new and effective heuristics.