Relaxed Implementation of spectral methods for graph partitioning

Abstract

This paper presents a fast implementation of the recursive spectral bisection method for p-way partitioning. It is known that recursive bisections for p-way partitioning using optimal strategies at each step may not lead to a good overall solution. The relaxed implementation accelerates the partitioning process by relaxing the accuracy requirement of spectral bisection (SB) method. Considering the solution quality of a SB method on a graph is primarily determined by the accuracy of its Fiedler vector, we propose to set a tight iteration number bound and a loose residual tolerance for Lanczos algorithms to compute the Fiedler vector. The relaxed SB was tested on eight representative meshes from different applications. Experimental results show that the relaxed SB on six meshes produces approximately equivalent quality solutions as the Chaco SB while gaining 10% to 35% improvements in execution time. On the other two meshes, the relaxed SB approaches the Chaco SB in quality as p increases and reduces the execution time by 40% to 70%. Coupled with the Kernighan-Lin local refinement algorithm, the relaxed SB is able to yield high quality solutions in all test cases as p goes beyond 32. Multilevel and spectral quadrisection algorithms benefit from relaxed implementations, as well.