New Abilities and Limitations of Spectral Graph Bisection

TL;DR

This paper analyzes Boppana's spectral graph bisection algorithm, demonstrating its effectiveness in semirandom and stochastic block models, and identifying its limitations when the graph parameters are too similar.

Contribution

It proves Boppana's algorithm performs optimally in semirandom and stochastic block models and establishes its equivalence with a recent SDP-based approach.

Findings

Boppana's algorithm works well in semirandom models.

The algorithm achieves optimal cluster recovery thresholds.

It fails when graph parameters are too close.

Abstract

Spectral based heuristics belong to well-known commonly used methods which determines provably minimal graph bisection or outputs "fail" when the optimality cannot be certified. In this paper we focus on Boppana's algorithm which belongs to one of the most prominent methods of this type. It is well known that the algorithm works well in the random \emph{planted bisection model} -- the standard class of graphs for analysis minimum bisection and relevant problems. In 2001 Feige and Kilian posed the question if Boppana's algorithm works well in the semirandom model by Blum and Spencer. In our paper we answer this question affirmatively. We show also that the algorithm achieves similar performance on graph classes which extend the semirandom model. Since the behavior of Boppana's algorithm on the semirandom graphs remained unknown, Feige and Kilian proposed a new semidefinite programming (SDP) based approach and proved that it works on this model. The relationship between the performance of the SDP based algorithm and Boppana's approach was left as an open problem. In this paper we solve the problem in a complete way by proving that the bisection algorithm of Feige and Kilian provides exactly the same results as Boppana's algorithm. As a consequence we get that Boppana's algorithm achieves the optimal threshold for exact cluster recovery in the \emph{stochastic block model}. On the other hand we prove some limitations of Boppana's approach: we show that if the density difference on the parameters of the planted bisection model is too small then the algorithm fails with high probability in the model.