On the Maximal Error of Spectral Approximation of Graph Bisection

TL;DR

This paper demonstrates that spectral graph bisection can significantly deviate from optimal solutions, especially on certain graph classes, with errors proportional to the square of the graph size.

Contribution

It provides a theoretical lower bound on the maximum error of spectral bisection methods for specific graph classes, highlighting limitations of the approach.

Findings

Spectral bisection can produce solutions far from optimal on certain graphs.

Maximum error in spectral approximation can be proportional to the square of the graph size.

The paper establishes a lower bound on spectral bisection accuracy for specific graph classes.

Abstract

Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error in the spectral approximation of the optimal bisection (partition sizes exactly equal) cut for such graphs is bounded below by a constant multiple of the order of the graph squared.