John C. Urschel

On the maximal error of spectral approximation of graph bisection

Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error in the spectral approximation of the optimal bisection (partition sizes exactly equal) cut for such graphs is bounded below by a constant multiple of the order of the graph squared.

On The Characterization and Uniqueness of Centroidal Voronoi Tessellations

Vector quantization is a classical signal-processing technique with significant applications in data compression, pattern recognition, clustering, and data stream mining. It is well known that for critical points of the quantization energy, the tessellation of the domain is a centroidal Voronoi tessellation. However, for dimensions greater than one, rigorously verifying a given centroidal Voronoi tessellation is a local minimum can prove difficult. Using variational techniques, we give a full characterization of the second variation of a centroidal Voronoi tessellation and give sufficient conditions for a centroidal Voronoi tessellation to be a local minimum. In addition, the conditions under which a centroidal Voronoi tessellation for a given density and domain is unique have been elusive for dimensions greater than one. We prove that there does not exist a unique two generator centroidal Voronoi tessellation for dimensions greater than one.