Expander flows, geometric embeddings and graph partitioning

Abstract

We give a O(sqrt(log n))-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in ℜ^d, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “certificate” for a graph’s expansion, by embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.