The minimum bisection in the planted bisection model

TL;DR

This paper derives an asymptotic formula for the minimum bisection width in the planted bisection model, revealing conditions under which the planted bisection is not minimal, thus advancing understanding of graph partitioning thresholds.

Contribution

It provides a new asymptotic formula for the minimum bisection width in the planted bisection model under specific parameter conditions.

Findings

Asymptotic formula for minimum bisection width derived

Planted bisection is not minimal under certain parameter regimes

Conditions involving $d_+$ and $d_-$ determine the bisection's optimality

Abstract

In the planted bisection model a random graph $G(n,p_+,p_- )$ with $n$ vertices is created by partitioning the vertices randomly into two classes of equal size (up to $\pm1$). Any two vertices that belong to the same class are linked by an edge with probability $p_+$ and any two that belong to different classes with probability $p_- <p_+$ independently. The planted bisection model has been used extensively to benchmark graph partitioning algorithms. If $p_{\pm} =2d_{\pm} /n$ for numbers $0\leq d_- <d_+ $ that remain fixed as $n\to\infty$, then w.h.p. the ``planted'' bisection (the one used to construct the graph) will not be a minimum bisection. In this paper we derive an asymptotic formula for the minimum bisection width under the assumption that $d_+ -d_- >c\sqrt{d_+ \ln d_+ }$ for a certain constant $c>0$.