Using markov-chain mixing time estimates for the analysis of ant colony optimization

The Markov chain Monte Carlo paradigm has developed powerful and elegant techniques for estimating the time until a Markov chain approaches a stationary distribution. This time is known as mixing time. We introduce the reader into mixing time estimations via coupling arguments and use the mixing of pheromone models for analyzing the expected optimization time of ant colony optimization. We demonstrate the approach for plateaus in pseudo-Boolean optimization and derive upper bounds for the time until a target set is found. We also describe how mixing times can be estimated for MMAS ant systems on shortest path problems.

[1]  M. Mitzenmacher,et al.  Probability and Computing: Chernoff Bounds , 2005 .

[2]  Santosh S. Vempala,et al.  Simulated annealing in convex bodies and an O*(n4) volume algorithm , 2006, J. Comput. Syst. Sci..

[3]  Alden H. Wright,et al.  Quotients of Markov chains and asymptotic properties of the stationary distribution of the Markov chain associated to an evolutionary algorithm , 2007, Genetic Programming and Evolvable Machines.

[4]  Dirk Sudholt,et al.  Analysis of different MMAS ACO algorithms on unimodal functions and plateaus , 2009, Swarm Intelligence.

[5]  Pietro Simone Oliveto,et al.  Theoretical analysis of fitness-proportional selection: landscapes and efficiency , 2009, GECCO.

[6]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[7]  Eric Vigoda,et al.  A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries , 2004, JACM.

[8]  Jittat Fakcharoenphol,et al.  A running time analysis of an Ant Colony Optimization algorithm for shortest paths in directed acyclic graphs , 2008, Inf. Process. Lett..

[9]  Thomas Jansen,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 Upper and Lower Bounds for Randomized Search Heuristics in Black-Box Optimization , 2004 .

[10]  Walter J. Gutjahr,et al.  First steps to the runtime complexity analysis of ant colony optimization , 2008, Comput. Oper. Res..

[11]  Frank Neumann,et al.  Theoretical Properties of Two ACO Approaches for the Traveling Salesman Problem , 2010, ANTS Conference.

[12]  Frank Neumann,et al.  Rigorous analyses of fitness-proportional selection for optimizing linear functions , 2008, GECCO '08.

[13]  Frank Neumann,et al.  Randomized Local Search, Evolutionary Algorithms, and the Minimum Spanning Tree Problem , 2004, GECCO.

[14]  Marc Schoenauer,et al.  Rigorous Hitting Times for Binary Mutations , 1999, Evolutionary Computation.

[15]  Benjamin Doerr,et al.  Refined runtime analysis of a basic ant colony optimization algorithm , 2007, 2007 IEEE Congress on Evolutionary Computation.

[16]  Thomas P. Hayes,et al.  Randomly coloring planar graphs with fewer colors than the maximum degree , 2007, STOC '07.

[17]  Dirk Sudholt,et al.  Ant colony optimization for stochastic shortest path problems , 2010, GECCO '10.

[18]  Dirk Sudholt,et al.  The benefit of migration in parallel evolutionary algorithms , 2010, GECCO '10.

[19]  Per Kristian Lehre,et al.  Ant colony optimization and the minimum cut problem , 2010, GECCO '10.

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

[21]  Dirk Sudholt,et al.  On the runtime analysis of the 1-ANT ACO algorithm , 2007, GECCO '07.

[22]  Carsten Witt,et al.  Rigorous Runtime Analysis of Swarm Intelligence Algorithms – An Overview , 2009 .

[23]  Jens Jägersküpper,et al.  When the Plus Strategy Outperforms the Comma Strategyand When Not , 2007, 2007 IEEE Symposium on Foundations of Computational Intelligence.

[24]  David J. Aldous,et al.  Lower bounds for covering times for reversible Markov chains and random walks on graphs , 1989 .

[25]  Dirk Sudholt,et al.  Running Time Analysis of ACO Systems for Shortest Path Problems , 2009, SLS.

[26]  Dirk Sudholt,et al.  Comparing Variants of MMAS ACO Algorithms on Pseudo-Boolean Functions , 2007, SLS.

[27]  Yuren Zhou,et al.  Runtime Analysis of an Ant Colony Optimization Algorithm for TSP Instances , 2009, IEEE Transactions on Evolutionary Computation.

[28]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[29]  Somenath Biswas,et al.  Necessary and sufficient conditions for success of the metropolis algorithm for optimization , 2010, GECCO '10.

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

[31]  B. Hajek.  Hitting-time and occupation-time bounds implied by drift analysis with applications , 1982, Advances in Applied Probability.

[32]  Frank Neumann,et al.  A rigorous view on neutrality , 2007, 2007 IEEE Congress on Evolutionary Computation.

[33]  Dirk Sudholt,et al.  Rigorous Analyses for the Combination of Ant Colony Optimization and Local Search , 2008, ANTS Conference.

[34]  Frank Neumann,et al.  Ant Colony Optimization and the minimum spanning tree problem , 2006, Theor. Comput. Sci..

[35]  Elizabeth L. Wilmer,et al.  Markov Chains and Mixing Times , 2008 .

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

[37]  Dirk Sudholt,et al.  A few ants are enough: ACO with iteration-best update , 2010, GECCO '10.

[38]  Frank Neumann,et al.  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 , 2022 .

[39]  W. Gutjahr,et al.  Runtime Analysis of Ant Colony Optimization with Best-So-Far Reinforcement , 2008 .