A running time analysis of an Ant Colony Optimization algorithm for shortest paths in directed acyclic graphs

In this paper, we prove polynomial running time bounds for an Ant Colony Optimization (ACO) algorithm for the single-destination shortest path problem on directed acyclic graphs. More specifically, we show that the expected number of iterations required for an ACO-based algorithm with n ants is O(1/ρn²m log n) for graphs with n nodes and m edges where ρ is an evaporation rate. This result can be modified to show that an ACO-based algorithm for One-Max with multiple ants converges in expected O(1/ρn² log n) iterations, where n is the number of variables. This result stands in sharp contrast with that of Neumann and Witt, where a single-ant algorithm is shown to require an exponential running time if ρ=O(n^−1−ϵ) for any ϵ>0.