Min-Max Graph Partitioning and Small Set Expansion

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O(root log n log k)-approximation algorithm. This improves over an O(log(2) n) approximation for the second version due to Svitkina and Tardos [22], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the Small-Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty set S subset of V of size vertical bar S vertical bar <= rho n with minimum edge-expansion. We give an O(root log n log (1/rho)) bicriteria approximation algorithm for the general case of Small-Set Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.