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

Matthias Kergaßner, Oliver Keszocze, Rolf Wanka

TL;DR

The paper investigates runtime analysis of Ant Colony Optimization by introducing Bivalent Ant Colony Optimization (BACO) with two pheromone values and a Markov-chain framework to obtain exact expected optimization times. It derives closed-form expressions for Leading-Ones, $T(n,t)=\frac{1+t}{2t^2}\cdot\big((1+t)^n-1\big)$, and a detailed formula for Sorting, uncovering tight bounds that yield $\Theta(n^2)$ for Leading-Ones with $t \approx 1/n$ and $\Theta(n^3)$ for Sorting with $t \approx 1/n^2$. The analysis also shows that the pheromone ratio $t$ critically controls performance, and the approach reproduces known $O(n\log n)$ results for OneMax as a by-product, with experiments validating the theoretical predictions. Overall, the work provides a general Markov-chain-based tool for analyzing ACO runtime and clarifies how pheromone discretization affects optimization time, offering a baseline for comparisons with other metaheuristics like the (1+1)-EA and OnePSO.

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 ($Θ(n^3)$) with a specifically chosen objective function and prove the missing lower bound $Ω(n^2)$ for LeadingOnes which, thus, is tightly bounded by $Θ(n^2)$. 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.

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

TL;DR

The paper investigates runtime analysis of Ant Colony Optimization by introducing Bivalent Ant Colony Optimization (BACO) with two pheromone values and a Markov-chain framework to obtain exact expected optimization times. It derives closed-form expressions for Leading-Ones, , and a detailed formula for Sorting, uncovering tight bounds that yield for Leading-Ones with and for Sorting with . The analysis also shows that the pheromone ratio critically controls performance, and the approach reproduces known results for OneMax as a by-product, with experiments validating the theoretical predictions. Overall, the work provides a general Markov-chain-based tool for analyzing ACO runtime and clarifies how pheromone discretization affects optimization time, offering a baseline for comparisons with other metaheuristics like the (1+1)-EA and OnePSO.

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 () with a specifically chosen objective function and prove the missing lower bound for LeadingOnes which, thus, is tightly bounded by . 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 () and LeadingOnes () can be re-produced as a by-product of our approach. Experiments validate our theoretical findings.
Paper Structure (9 sections, 8 theorems, 34 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 9 sections, 8 theorems, 34 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Bivalent Ant Colony Optimization
  3. Markov Chain-based Optimization Time Analysis of BACO
  4. LeadingOnes
  5. Sorting
  6. Discussion of the Results
  7. Experiments
  8. Conclusion & Outlook
  9. Infinite sum of matrix powers

Key Result

Theorem 3.1

Let $p$ be the probability distribution vector of the initial Markov state and $M$ the Markov matrix. The expected optimization time $T(n,t)$ of BACO with the pheromone ratio $t$ is given by where $\mathbb{1}$ denotes the unit matrix and the vector $(0,\dots,0,1)^\top$ consists of zeros and a one as the last entry, both of fitting dimension.

Figures (2)

  • Figure 1: Optimization time for Leading-Ones and the analytically calculated expected optimization time using the pheromone ratio $t=\frac{1}{n}$.
  • Figure 2: Optimization time for Sorting and the analytically calculated expected optimization time using the pheromone ratio $t=\frac{1}{n^2}$.

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.6
  • Theorem 3.7
  • proof
  • Lemma 3.10
  • ...and 5 more