Towards an SDP-based Approach to Spectral Methods: A Nearly-Linear-Time Algorithm for Graph Partitioning and Decomposition

TL;DR

This paper introduces a nearly-linear-time spectral algorithm for graph partitioning that improves efficiency and guarantees optimal approximation, using a novel SDP approach and a simple, conceptually clear method.

Contribution

It presents the first nearly-linear-time spectral algorithm for the Balanced Separator problem with optimal approximation guarantees, utilizing a new SDP-based approach and a novel separation oracle.

Findings

Achieves nearly-linear time complexity of O(|E|/)

Provides the first spectral method with optimal approximation guarantees

Introduces a novel SDP separation oracle for graph partitioning

Abstract

In this paper, we consider the following graph partitioning problem: The input is an undirected graph $G=(V,E),$ a balance parameter $b \in (0,1/2]$ and a target conductance value $\gamma \in (0,1).$ The output is a cut which, if non-empty, is of conductance at most $O(f),$ for some function $f(G, \gamma),$ and which is either balanced or well correlated with all cuts of conductance at most $\gamma.$ Spielman and Teng gave an $\tilde{O}(|E|/\gamma^{2})$-time algorithm for $f= \sqrt{\gamma \log^{3}|V|}$ and used it to decompose graphs into a collection of near-expanders. We present a new spectral algorithm for this problem which runs in time $\tilde{O}(|E|/\gamma)$ for $f=\sqrt{\gamma}.$ Our result yields the first nearly-linear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primal-dual semidefinite-programming SDP approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than $\gamma,$ somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale's framework to bound the running time. Our result contains technical ingredients that may be of independent interest.