Graph bisection revisited

TL;DR

This paper introduces a simplified semidefinite programming relaxation for the graph bisection problem, which is computationally more efficient and provides stronger bounds than previous methods.

Contribution

A new, smaller semidefinite relaxation for graph bisection that allows for additional inequalities, improving solution bounds.

Findings

Provides the strongest bounds for graph bisection to date

Reduces computational complexity compared to previous relaxations

Numerical results confirm improved performance

Abstract

The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation for the graph bisection problem with a matrix variable of order $n$ - the number of vertices of the graph - that is equivalent to the currently strongest semidefinite programming relaxation obtained by using vector lifting. The reduction in the size of the matrix variable enables us to impose additional valid inequalities to the relaxation in order to further strengthen it. The numerical results confirm that our simplified and strengthened semidefinite relaxation provides the currently strongest bound for the graph bisection problem in reasonable time.