Abstract
Recently, the first rigorous runtime analyses of ACO algorithms appeared, covering variants of the MAX–MIN ant system and their runtime on pseudo-Boolean functions. Interestingly, a variant called 1-ANT is very sensitive to the evaporation factor while Gutjahr and Sebastiani proved partly opposite results for their variant MMASbs. These algorithms differ in their pheromone update mechanisms and, moreover, 1-ANT accepts equally fit solutions in contrast to MMASbs.
By analyzing variants of MMASbs, we prove that the different behavior of 1-ANT and MMASbs results from the different pheromone update mechanisms. Building upon results by Gutjahr and Sebastiani, we extend their analyses of MMASbs to the class of unimodal functions and show improved results for test functions using new and specialized techniques; in particular, we present new lower bounds. Finally, we compare MMASbs with a variant that also accepts equally fit solutions as this enables the exploration of plateaus. For well-known plateau functions we prove that this drastically reduces the optimization time. Our findings are complemented by experiments that support our asymptotic analyses and yield additional insights.
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References
Brémaud, P. (1998). Markov chains: Gibbs fields, Monte Carlo simulation, and queues. New York: Springer.
Doerr, B., & Johannsen, D. (2007). Refined runtime analysis of a basic ant colony optimization algorithm. In Proceedings of the congress on evolutionary computation (CEC ’07) (pp. 501–507). New York: IEEE Press.
Doerr, B., Hebbinghaus, N., & Neumann, F. (2006). Speeding up evolutionary algorithms through restricted mutation operators. In LNCS : Vol. 4193. Parallel problem solving from nature (PPSN ’06) (pp. 978–987). Berlin: Springer.
Doerr, B., Neumann, F., Sudholt, D., & Witt, C. (2007). On the runtime analysis of the 1-ANT ACO algorithm. In Proceedings of the genetic and evolutionary computation conference (GECCO ’07) (pp. 33–40). New York: ACM.
Dorigo, M., & Blum, C. (2005). Ant colony optimization theory: A survey. Theoretical Computer Science, 344, 243–278.
Dorigo, M., & Stützle, T. (2004). Ant colony optimization. Cambridge: MIT Press.
Dorigo, M., Maniezzo, V., & Colorni, A. (1996). Ant system: Optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics Part B, 26(1), 29–41.
Droste, S., Jansen, T., & Wegener, I. (2002). On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science, 276, 51–81.
Droste, S., Jansen, T., & Wegener, I. (2006). Upper and lower bounds for randomized search heuristics in black-box optimization. Theory of Computing Systems, 39(4), 525–544.
Eiben, A., & Smith, J. (2007). Introduction to evolutionary computing (2nd ed.). Berlin: Springer.
Garnier, J., Kallel, L., & Schoenauer, M. (1999). Rigorous hitting times for binary mutations. Evolutionary Computation, 7(2), 173–203.
Giel, O., & Wegener, I. (2003). Evolutionary algorithms and the maximum matching problem. In LNCS : Vol. 2607. Proceedings of the 20th annual symposium on theoretical aspects of computer science (STACS ’03) (pp. 415–426). Berlin: Springer.
Gutjahr, W. J. (2002). ACO algorithms with guaranteed convergence to the optimal solution. Information Processing Letters, 82(3), 145–153.
Gutjahr, W. J. (2006). On the finite-time dynamics of ant colony optimization. Methodology and Computing in Applied Probability, 8(1), 105–133.
Gutjahr, W. J. (2007). Mathematical runtime analysis of ACO algorithms: Survey on an emerging issue. Swarm Intelligence, 1, 59–79.
Gutjahr, W. J. (2008). First steps to the runtime complexity analysis of ant colony optimization. Computers and Operations Research, 35(9), 2711–2727.
Gutjahr, W. J., & Sebastiani, G. (2008). Runtime analysis of ant colony optimization with best-so-far reinforcement. Methodology and Computing in Applied Probability, 10, 409–433.
Jansen, T. Sudholt, D. (2009, to appear). Analysis of an asymmetric mutation operator. Evolutionary Computation.
Jansen, T., & Wegener, I. (2001). Evolutionary algorithms—how to cope with plateaus of constant fitness and when to reject strings of the same fitness. IEEE Transactions on Evolutionary Computation, 5(6), 589–599.
Mitzenmacher, M., & Upfal, E. (2005). Probability and computing—randomized algorithms and probabilistic analysis. Cambridge: Cambridge University Press.
Motwani, R., & Raghavan, P. (1995). Randomized algorithms. Cambridge: Cambridge University Press.
Neumann, F. (2004). Expected runtimes of evolutionary algorithms for the Eulerian cycle problem. In Proceedings of the congress on evolutionary computation (CEC ’04) (Vol. 1, pp. 904–910). New York: IEEE Press.
Neumann, F. (2007). Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem. European Journal of Operational Research, 181(3), 1620–1629.
Neumann, F., & Wegener, I. (2006). Minimum spanning trees made easier via multi-objective optimization. Natural Computing, 5(3), 305–319.
Neumann, F., & Wegener, I. (2007). Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theoretical Computer Science, 378(1), 32–40.
Neumann, F., & Witt, C. (2006). Runtime analysis of a simple ant colony optimization algorithm. In LNCS : Vol. 4288. Proceedings of the 17th international symposium on algorithms and computation (ISAAC ’06) (pp. 618–627). Berlin: Springer. Extended version to appear in Algorithmica.
Neumann, F., Sudholt, D., & Witt, C. (2007). Comparing variants of MMAS ACO algorithms on pseudo-Boolean functions. In LNCS : Vol. 4638. Proceedings of engineering stochastic local search algorithms (SLS ’07) (pp. 61–75). Berlin: Springer.
Rudolph, G. (1997). Convergence properties of evolutionary algorithms. Hamburg: Kovač.
Scharnow, J., Tinnefeld, K., & Wegener, I. (2004). The analysis of evolutionary algorithms on sorting and shortest paths problems. Journal of Mathematical Modelling and Algorithms, 3(4), 349–366.
Stützle, T., & Hoos, H. H. (2000). MAX-MIN ant system. Future Generations Computer Systems, 16, 889–914.
Wegener, I. (2002). Methods for the analysis of evolutionary algorithms on pseudo-Boolean functions. In R. Sarker, X. Yao, & M. Mohammadian (Eds.), Evolutionary optimization (pp. 349–369). Dordrecht: Kluwer Academic.
Wegener, I., & Witt, C. (2005). On the optimization of monotone polynomials by simple randomized search heuristics. Combinatorics, Probability and Computing, 14, 225–247.
Witt, C. (2005). Worst-case and average-case approximations by simple randomized search heuristics. In LNCS : Vol. 3404. Proceedings of the 22nd symposium on theoretical aspects of computer science (STACS ’05) (pp. 44–56). Berlin: Springer.
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A conference version appeared in SLS 2007 (Neumann et al. 2007).
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Neumann, F., Sudholt, D. & Witt, C. Analysis of different MMAS ACO algorithms on unimodal functions and plateaus. Swarm Intell 3, 35–68 (2009). https://doi.org/10.1007/s11721-008-0023-3
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DOI: https://doi.org/10.1007/s11721-008-0023-3