First steps to the runtime complexity analysis of ant colony optimization

The paper presents results on the runtime complexity of two ant colony optimization (ACO) algorithms: ant system, the oldest ACO variant, and GBAS, the first ACO variant for which theoretical convergence results have been established. In both cases, as the class of test problems under consideration, a slight generalization of the well-known OneMax test function has been chosen. The techniques used for the runtime analysis of the two algorithms differ: in the case of GBAS, the expected runtime until the optimal solution is reached is studied by a direct bound estimation approach inspired by comparable results for the (1+1)$(1+1)$ evolutionary algorithm (EA). A runtime bound of order O(mlogm)$O(m\log m)$, where m   is the problem instance size, is obtained. In the case of ant system, the original discrete stochastic process is approximated by a suitable continuous deterministic process. The validity of the approximation is shown by means of a rigid convergence theorem exploiting a classical result from mathematical learning theory. Using this approximation, it is demonstrated that for the considered OneMax-type problems, a runtime of order O(mlog(1/ε))$O(m\log (1/\epsilon ))$ until reaching an expected relative   solution quality of 1-ε$1-\epsilon$, and a runtime of O(mlogm)$O(m\log m)$ until reaching the optimal   solution with high probability can be predicted. Our results are the first to show competitiveness in runtime complexity with (1+1$1+1$) EA on OneMax for a proper ACO algorithm.