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.

翻译：迄今为止，仅有少数关于蚁群优化（ACO）运行时行为的上界被报道。为改善这一现状，我们研究了一种称为二值蚁群优化（BACO）的ACO变体，其仅使用两种信息素值。我们提出并成功应用了一种基于马尔可夫链的新方法，用于计算期望优化时间，即算法终止前的预期迭代次数。该方法能够推导出排序（Sorting）和LeadingOnes问题在期望优化时间上的精确公式。结果表明，两种信息素值的比例显著决定了BACO的运行时行为。据我们所知，我们首次针对具有特定目标函数的排序问题给出了紧界（$\Theta(n^3)$），并证明了LeadingOnes问题缺失的下界$\Omega(n^2)$，由此其紧界为$\Theta(n^2)$。我们证明，尽管在信息素对求解过程的影响方面，该蚁群算法被大幅简化，但我们的方法仍能作为副产品复现已知的OneMax问题（$O(n\log n)$）和LeadingOnes问题（$O(n^2)$）的期望优化时间上界。实验验证了我们的理论发现。