Spectral Analysis for Graph Partitioning

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Graph partitioning is a fundamental combinatorial optimizationproblem that has many practical applications such asin supporting efficient load balancing for parallel processing,in VLSI layout, and in data clustering. This proposed research program focuses on the study of spectral methods for graph partitioning.Spectral methods make use of the eigenvectors of graph matrices (e.g., the Laplacian or the adjacency matrix of a graph) to construct a qualitypartitioning. They have been popularly used in practicefor partitioning meshes in scientific simulation, for dividing graphsderived from circuits, and for clustering data in web-graph analysisand information organization. However, the quality of the partition that these methods should produce has so far eluded precise analysis.Spielman and the PI made some breakthrough progresses.In particular, by proving that the second smallest eigenvalue ofthe Laplacian matrices of bounded-degree planar graphsis at most O(1/n), Spielman and the PIshowed that proper use of spectral techniques can producea bisection of graphs with cut size at most $O(\sqrt{n})$,which is best possible for the family of planar graphs.