New heuristics and lower bounds for graph partitioning

Abstract

Addresses the problem of partitioning the vertex set of an edge-weighted undirected graph into two parts of specified sizes so that the sum of the weights on edges joining vertices in different parts is minimum. The authors report on a new class of algorithms that solve the graph partitioning problem in polynomial time by using low-rank approximations of the connection matrix obtained from principal components analysis. These algorithms also provide a bound on the proximity of the cost of the constructed partition to the optimal cost based on the eigenvalues left out in the rank reduction process. The lower bounds derived are proven to be superior to the Donath-Hoffman lower bound for two significant special cases, wherein either the two part sizes are equal, or the graph connection matrix has equal row-sums. Simulation results for randomly constructed graphs of different sizes are presented. >