A Multilevel Algorithm for Spectral Partitioning with Extended Eigen-Models Suely Oliveira 1 and Takako Soma 1 The Department of Computer Science, The University of Iowa, Iowa City, IA 52242, USA, oliveira@cs.uiowa.edu, WWW home page: http://www.cs.uiowa.edu/~oliveira Abstract. Parallel solution of irregular problems require solving the graph partitioning problem. The extended eigenproblem appears as the solution of some relaxed formulations of the graph partitioning problem. In this paper, a new subspace algorithm for the solving the extended eigenproblem is presented. The structure of this subspace method allows the incorporation of multigrid preconditioners. We numerically compare our new algorithm with a previous algorithm based on Lanczos iteration and show that our subspace algorithm performs better. 1 Introduction One of the main problems encountered when dealing with irregular problems on parallel architectures is mapping the data into the various processors. Tradition- ally graph partitioning has been used to achieve this goal. Kernighan and Lin developed an effective combinatorial method based on swapping vertices [10]. Multilevel extensions of the Kernighan-Lin algorithm have proven effective for graphs with large numbers of vertices [9]. Like combi- natorial methods, spectral methods have proven effective for large graphs aris- ing from FEM discretizations [12]. In fact, currently various software packages (METIS [8] and Chaco [4] and others) combine multilevel combinatorial algo- rithms and spectral algorithms. The spectral algorithms which are used in these packages are based on the models that partition a graph by finding the second smallest eigenvector of its graph Laplacian using an iterative method. This is in fact the model that we used in [7] and [6]. However, as the example below shows, there are number of reasons for mod- ifying the traditional model and spectral heuristics for graph partitioning. Re- cently, Hendrickson et al. [5] pointed out the problems with traditional models. Consider Figure 1. Assume we have already partitioned the graph into two pieces, left and right halves, and that we have similarly divided the left half graph into top and bottom quadrants. When partitioning the right half graph between pro- cessors 3 and 4 we should like messages to travel short distances. The mapping shown in the left-hand figure is better since the total message distance is less than that for the right-hand figure. Note that even though in modern computers