Approximation Algorithms for Finding Maximum Induced Expanders

TL;DR

This paper introduces a novel approximation algorithm for finding large induced expanders in graphs, utilizing a new semidefinite programming relaxation, with potential applications in graph partitioning and the unique games problem.

Contribution

The paper presents the first bi-criteria approximation algorithm for the maximum induced expander problem using a novel SDP relaxation.

Findings

Provides a bi-criteria approximation with explicit bounds

Introduces a new SDP relaxation for the problem

Suggests applications to graph partitioning and the unique games problem

Abstract

We initiate the study of approximating the largest induced expander in a given graph $G$. Given a $\Delta$-regular graph $G$ with $n$ vertices, the goal is to find the set with the largest induced expansion of size at least $\delta \cdot n$. We design a bi-criteria approximation algorithm for this problem; if the optimum has induced spectral expansion $\lambda$ our algorithm returns a $\frac{\lambda}{\log^2\delta \exp(\Delta/\lambda)}$-(spectral) expander of size at least $\delta n$ (up to constants). Our proof introduces and employs a novel semidefinite programming relaxation for the largest induced expander problem. We expect to see further applications of our SDP relaxation in graph partitioning problems. In particular, because of the close connection to the small set expansion problem, one may be able to obtain new insights into the unique games problem.