Refined runtime analysis of a basic ant colony optimization algorithm

Abstract

Neumann and analyzed the runtime of the basic ant colony optimization (ACO) algorithm 1-Ant on pseudoboolean optimization problems. For the problem OneMax they showed how the runtime depends on the evaporation factor. In particular, they proved a phase transition from exponential to polynomial runtime. In this work, we simplify the view on this problem by an appropriate translation of the pheromone model. This results in a profound simplification of the pheromone update rule and, by that, a refinement of the results of Neumann and Witt. In particular, we show how the exponential runtime bound gradually changes to a polynomial bound inside the phase of transition.

How Single Ant ACO Systems Optimize Pseudo-Boolean Functions

“…In [11] it was shown that the expected optimization time of 1-Ant on OneMax is polynomial if ρ = 1 − 1/n with fixed > 0 and in [2] that for ρ = o(1/ log n) it becomes super-polynomial. On the function LeadingOnes the expected optimization time was shown [3] to be quadratic for constant ρ, polynomial for ρ = Ω(1/ log n), and again super-polynomial for ρ = o(1/ log n).…”

“…Additionally, it was observed that for ρ very close to one, 1-Ant exactly simulates the (1+1) EA, which was rigorously analyzed in [6]. In [2] the pheromone model used in [11] was replaced by a simpler, but equivalent model, in which the pheromone values equal the probabilities of ants walking along a particular edge in the construction graph. In [3] an analysis of 1-Ant was carried out for the functions LeadingOnes and BinaryValue.…”

“…To gain an understanding how these algorithms work, we track a number of theory-guided indicators (other than the resulting optimization time) during the runs of 1-Ant [11] and the MMAS [12]. For both algorithms we use the pheromone system described in [2] which is equivalent to the one used in [8], [11], and [3].…”

Comparison of different Ant based techniques for identification of shortest path in Distributed Network

“…In 2006 by Neumann and Witt [2], Doerr, Neumann, Sudholt, and in 2007 by Witt [10], and Doerr and Johannsen [11] studied a simple algorithm 1-ANT that constructs a pseudo Boolean solution according to a straightforward construction graph where an ant makes independent choices for each bit. The 1-ANT records the best solution found so far.…”

Ant Colony Optimization and hypergraph covering problems

“…Investigations on the runtime performance of ACO first started for various pseudo-boolean functions [3,10,11,12] where a simple ACO variant 1-ANT was analysed like (1+1) EA [7]. Another ACO variant MMAS has also been widely investigated for which the phase transition present in 1-ANT due to the evaporation factor does not occur [13,14].…”