Markov Chain-based Optimization Time Analysis of Bivalent Ant Colony Optimization for Sorting and LeadingOnes

TL;DR

This paper introduces a Markov chain-based method to analyze the expected runtime of a simplified Bivalent Ant Colony Optimization algorithm on sorting and LeadingOnes problems, deriving tight bounds and validating results experimentally.

Contribution

It provides the first exact formulas and tight bounds for BACO's runtime on specific problems, revealing the influence of pheromone ratios and simplifying previous analyses.

Findings

Expected optimization time for Sorting is Θ(n^3).

Expected optimization time for LeadingOnes is Θ(n^2).

Known bounds for OneMax and LeadingOnes are confirmed through the new approach.

Abstract

So far, only few bounds on the runtime behavior of Ant Colony Optimization (ACO) have been reported. To alleviate this situation, we investigate the ACO variant we call Bivalent ACO (BACO) that uses exactly two pheromone values. We provide and successfully apply a new Markov chain-based approach to calculate the expected optimization time, i. e., the expected number of iterations until the algorithm terminates. This approach allows to derive exact formulae for the expected optimization time for the problems Sorting and LeadingOnes. It turns out that the ratio of the two pheromone values significantly governs the runtime behavior of BACO. To the best of our knowledge, for the first time, we can present tight bounds for Sorting ($\Theta(n^3)$) with a specifically chosen objective function and prove the missing lower bound $\Omega(n^2)$ for LeadingOnes which, thus, is tightly bounded by $\Theta(n^2)$. We show that despite we have a drastically simplified ant algorithm with respect to the influence of the pheromones on the solving process, known bounds on the expected optimization time for the problems OneMax ($O(n\log n)$) and LeadingOnes ($O(n^2)$) can be re-produced as a by-product of our approach. Experiments validate our theoretical findings.