Almost Optimal Local Graph Clustering Using Evolving Sets

Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A <i>local graph partitioning algorithm</i> finds a set of vertices with small conductance (i.e., a sparse cut) by adaptively exploring part of a large graph <i>G</i>, 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 <i>n</i> of vertices in <i>G</i>. Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank [Spielman and Teng 2013; Andersen et al. 2006]. In this article, we introduce a simple randomized local partitioning algorithm that finds a sparse cut by simulating the <i>volume-biased evolving set process</i>, which is a Markov chain on sets of vertices. We prove that for any ε > 0, and any set of vertices <i>A</i> that has conductance at most φ, for at least half of the starting vertices in <i>A</i> our algorithm will output (with constant probability) a set of conductance <i>O</i>(&sqrt;φ /ε). 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 vol(<i>A</i>)^εφ^&mins;1/2polylog(<i>n</i>), where vol(<i>A</i>) = ∑_{<i>v</i> ∈ <i>A</i>}<i>d</i>(<i>v</i>) is the volume of the set <i>A</i>. This gives an algorithm with the same guarantee (up to a constant factor) as the Cheeger’s inequality that runs in time slightly superlinear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheeger’s inequality. In comparison, the best previous local partitioning algorithm, by Andersen et al. [2006], has a worse approximation guarantee of <i>O</i>(&sqrt;φ log <i>n</i>) and a larger ratio of φ^&mins;1 polylog(<i>n</i>) between the complexity and output volume. As a by-product of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. For 0 < <i>k</i> ⩽ vol(<i>V</i>)/2, let φ(<i>k</i>) ≔ min _{<i>S</i>: vol(<i>S</i>) ⩽ <i>k</i>}φ(<i>S</i>). There is a polynomial time algorithm that, for any <i>k</i>, ε > 0, finds a set <i>S</i> of volume vol(<i>S</i>) ⩽ <i>O</i>(<i>k</i>^{1 + ε}) and expansion φ(<i>S</i>)≤ <i>O</i>(&sqrt;φ (<i>k</i>)/ε). As a new technical tool, we show that for any set <i>S</i> of vertices of a graph, a lazy <i>t</i>-step random walk started from a randomly chosen vertex of <i>S</i> will remain entirely inside <i>S</i> with probability at least (1 − φ(<i>S</i>)/2)^<i>t</i>. This itself provides a new lower bound to the uniform mixing time of any finite state reversible Markov chain.