Fast approximate graph partitioning algorithms

We study graph partitioning problems on graphs with edge capacities and vertex weights. The problems of b-balanced cuts and k-balanced partitions are unified into a new problem called minimum capacity rho-separators. A rho-separator is a subset of edges whose removal partitions the vertex set into connected components such that the sum of the vertex weights in each component is at most rho times the weight of the graph. We present a new and simple O(log n)-approximation algorithm for minimum capacity rho-separators which is based on spreading metrics yielding an O(log n)-approximation algorithm both for b-balanced cuts and k-balanced partitions. In particular, this result improves the previous best known approximation factor for k-balanced partitions in undirected graphs by a factor of O(log k). We enhance these results by presenting a version of the algorithm that obtains an O(log OPT)-approximation factor. The algorithm is based on a technique called spreading metrics that enables us to formulate directly the minimum capacity rho-separator problem as an integer program. We also introduce a generalization called the simultaneous separator problem, where the goal is to find a minimum capacity subset of edges that separates a given collection of subsets simultaneously. We extend our results to directed graphs for values of rho greater than or equal to 1/2. We conclude with an efficient algorithm for computing an optimal spreading metric for rho-separators. This yields more efficient algorithms for computing b-balanced cuts than were previously known.