Expander flows, geometric embeddings and graph partitioning

Abstract

We give a O (√log n )-approximation algorithm for the sparsest cut, edge expansion, 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 “approximate certificate” for a graph's expansion, which involves embedding an n -node expander in it with appropriate dilation and congestion. We call this an expander flow.