Finding sparse cuts locally using evolving sets
Reid Andersen, Yuval Peres · 2009
A local graph partitioning algorithm finds a set of vertices with small conductance (i.e. a sparse cut) by adaptively exploring part of a large graph G, starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number n of vertices in G. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank. In this paper, we introduce a randomized local partitioning algorithm that finds a sparse cut by simulating the volume-biased evolving set process, which is a Markov chain on sets of vertices. We prove that for any set of vertices A that has conductance at most φ, for at least half of the starting vertices in A our algorithm will output (with probability at least half), a set of conductance O(φ 1/2 log 1/2 n). We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is O(φ −1/2 polylog(n)). In comparison, the best previous local partitioning algorithm, due to Andersen, Chung, and Lang, has the same approximation guarantee, but a larger ratio of O(φ −1 polylog(n)) between the complexity and output volume. Using our local partitioning algorithm as a subroutine, we construct a fast algorithm for finding balanced cuts. Given a fixed value of φ, the resulting algorithm has complexity (m + nφ −1/2)) · O(polylog(n)) and returns a cut with conductance O(φ 1/2 log 1/2 n) and volume at least vφ/2, where vφ is the largest volume of any set with conductance at most φ. 1 1