Simple Max-Min Ant Systems and the Optimization of Linear Pseudo-Boolean Functions

TL;DR

This paper analyzes the runtime of simple MAX-MIN ant systems on linear pseudo-Boolean functions, providing theoretical bounds and experimental insights into their optimization behavior.

Contribution

It offers the first formal analysis of MAX-MIN ant systems on linear pseudo-Boolean functions, including upper bounds and improved results for specific functions.

Findings

General upper bound of O((n^3 log n)/ρ) for all linear functions

Improved bounds for OneMax and BinVal functions

Experimental results supporting theoretical analysis

Abstract

With this paper, we contribute to the understanding of ant colony optimization (ACO) algorithms by formally analyzing their runtime behavior. We study simple MAX-MIN ant systems on the class of linear pseudo-Boolean functions defined on binary strings of length 'n'. Our investigations point out how the progress according to function values is stored in pheromone. We provide a general upper bound of O((n^3 \log n)/ \rho) for two ACO variants on all linear functions, where (\rho) determines the pheromone update strength. Furthermore, we show improved bounds for two well-known linear pseudo-Boolean functions called OneMax and BinVal and give additional insights using an experimental study.