 
Summary: Expander Flows, Geometric Embeddings and Graph Partitioning
(Longer version of an ACM STOC 2004 paper)
Sanjeev Arora
Satish Rao
Umesh Vazirani
April 2, 2007
Abstract
We give a O(
log n)approximation algorithm for sparsest cut, edge expansion, balanced sepa
rator, and graph conductance problems. This improves the O(log n)approximation of Leighton and
Rao (1988). We use a wellknown semidefinite relaxation with triangle inequality constraints. Central to
our analysis is a geometric theorem about projections of point sets in 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 nnode expander in it with appropriate dilation and congestion. We call this an
expander flow.
1 Introduction
Partitioning a graph into two (or more) large pieces while minimizing the size of the "interface" between
