Cut Size Statistics of Graph Bisection Heuristics

TL;DR

This paper studies the statistical distribution of cut sizes produced by heuristic algorithms for graph bisection on sparse random graphs, revealing Gaussian tendencies and proposing a ranking method based on solution quality and speed.

Contribution

It empirically characterizes the distribution of cut sizes and introduces a ranking procedure for heuristics considering both quality and efficiency.

Findings

Distribution of cut sizes tends to Gaussian as graph size increases

Mean and variance of cut sizes scale linearly with number of vertices

Proposed ranking method balances solution quality and algorithm speed

Abstract

We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve approximately the graph bisection problem. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by ``local'' algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends towards a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given the distribution of cut sizes associated with each heuristic, we provide a ranking procedure which takes into account both the quality of the solutions and the speed of the algorithms. This procedure is demonstrated for a selection of local graph bisection heuristics.