Improved Cheeger's Inequality and Analysis of Local Graph Partitioning using Vertex Expansion and Expansion Profile

Abstract

We prove two generalizations of the Cheeger's inequality. The first generalization relates the second eigenvalue to the edge expansion and the vertex expansion of the graph $G$, $\lambda_2 = \Omega( \phi^V(G) \phi(G) )$, where $\phi^V(G)$ denotes the robust vertex expansion of $G$ and $\phi(G)$ denotes the edge expansion of $G$. The second generalization relates the second eigenvalue to the edge expansion and the expansion profile of $G$, for all $k \geq 2$, $ \lambda_2 = \Omega( \phi_k(G) \phi(G) / k )$, where $\phi_k(G)$ denotes the $k$-way expansion of $G$. These show that the spectral partitioning algorithm has better performance guarantees when $\phi^V(G)$ is large (e.g., planted random instances) or $\phi_k(G)$ is large (instances with few disjoint nonexpanding sets). Both bounds are tight up to a constant factor. Our approach is based on a method to analyze solutions of Laplacian systems, and this allows us to extend the results to local graph partitioning algorithms. In particular, we show that ou...