Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods Runtime Analysis of a Simple Ant Colony Optimization Algorithm Runtime Analysis of a Simple Ant Colony Optimization Algorithm

Abstract Ant Colony Optimization (ACO) has become quite popular in recent years. In contrast to many successful applications, the theoretical foundation of this randomized search heuristic is rather weak. Building up such a theory is demanded to understand how these heuristics work as well as to come up with better algorithms for certain problems. Up to now, only convergence results have been achieved showing that optimal solutions can be obtained in finite time. We present the first runtime analysis of an ACO algorithm, which transfers many rigorous results with respect to the runtime of a simple evolutionary algorithm to our algorithm. Moreover, we examine the choice of the evaporation factor, a crucial parameter in ACO algorithms, in detail. By deriving new lower bounds on the tails of sums of independent Poisson trials, we determine the effect of the evaporation factor almost completely and prove a phase transition from exponential to polynomial runtime.

[1]  L. Dworsky.  An Introduction to Probability , 2008 .

[2]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[3]  Ingo Wegener,et al.  Simulated Annealing Beats Metropolis in Combinatorial Optimization , 2005, ICALP.

[4]  C. Scheideler.  Probabilistic Methods for Coordination Problems , 2000 .

[5]  Martin Middendorf,et al.  Modeling the Dynamics of Ant Colony Optimization , 2002, Evolutionary Computation.

[6]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[7]  Mark Jerrum,et al.  Simulated annealing for graph bisection , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[8]  W. Hoeffding.  On the Distribution of the Number of Successes in Independent Trials , 1956 .

[9]  M. Dorigo,et al.  Ant System: An Autocatalytic Optimizing Process , 1991 .

[10]  J. Darroch.  On the Distribution of the Number of Successes in Independent Trials , 1964 .

[11]  Charles M. Grinstead,et al.  Introduction to probability , 1986, Statistics for the Behavioural Sciences.

[12]  Marco Dorigo,et al.  Ant colony optimization theory: A survey , 2005, Theor. Comput. Sci..

[13]  Ingo Wegener,et al.  Randomized local search, evolutionary algorithms, and the minimum spanning tree problem , 2004, Theor. Comput. Sci..

[14]  Mark Jerrum,et al.  The Metropolis Algorithm for Graph Bisection , 1998, Discret. Appl. Math..

[15]  Ingo Wegener,et al.  Evolutionary Algorithms and the Maximum Matching Problem , 2003, STACS.

[16]  Carsten Witt,et al.  UNIVERSITY OF DORTMUND REIHE COMPUTATIONAL INTELLIGENCE COLLABORATIVE RESEARCH CENTER 531 Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Worst-Case and Average-Case Approximations by Simple Randomized Search Heuristics , 2004 .

[17]  Frank Neumann,et al.  Runtime Analysis of a Simple Ant Colony Optimization Algorithm , 2006, ISAAC.

[18]  Mihalis Yannakakis,et al.  On the complexity of local search , 1990, STOC '90.

[19]  W. Gutjahr.  A GENERALIZED CONVERGENCE RESULT FOR THE GRAPH-BASED ANT SYSTEM METAHEURISTIC , 2003, Probability in the Engineering and Informational Sciences.

[20]  Thomas Jansen,et al.  and Management , 1998 .

[21]  Costas S. Iliopoulos,et al.  Symposium on Theoretical Aspects of Computer Science , 2008 .

[22]  Thomas Stützle,et al.  Ant Colony Optimization Theory , 2004 .

[23]  W. Gutjahr.  On the Finite-Time Dynamics of Ant Colony Optimization , 2006 .