Performance Analysis of Evolutionary Algorithms for the Minimum Label Spanning Tree Problem

A few experimental investigations have shown that evolutionary algorithms (EAs) are efficient for the minimum label spanning tree (MLST) problem. However, we know little about that in theory. In this paper, we theoretically analyze the performances of the (1+1) EA, a simple version of EA, and a simple multiobjective evolutionary algorithm called GSEMO on the MLST problem. We reveal that for the MLSTb problem, the (1+1) EA and GSEMO achieve a (b + 1)/2-approximation ratio in expected polynomial runtime with respect to n, the number of nodes, and k, the number of labels. We also find that GSEMO achieves a (2 lnn+1)-approximation ratio for the MLST problem in expected polynomial runtime with respect to n and k. At the same time, we show that the (1+1) EA and GSEMO outperform local search algorithms on three instances of the MLST problem. We also construct an instance on which GSEMO outperforms the (1+1) EA.

[1]  Sergio Consoli,et al.  Solving the minimum labelling spanning tree problem using hybrid local search , 2012, Electron. Notes Discret. Math..

[2]  Xin Yao,et al.  On the approximation ability of evolutionary optimization with application to minimum set cover , 2010, Artif. Intell..

[3]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[4]  Xin Yao,et al.  A New Approach for Analyzing Average Time Complexity of Population-Based Evolutionary Algorithms on Unimodal Problems , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[5]  Guoliang Chen,et al.  A note on the minimum label spanning tree , 2002, Inf. Process. Lett..

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

[7]  Günther R. Raidl,et al.  Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques , 2011, Adv. Oper. Res..

[8]  Günther R. Raidl,et al.  Compressing Fingerprint Templates by Solving an Extended Minimum Label Spanning Tree Problem , 2007 .

[9]  Kenneth A. De Jong,et al.  Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods on the Choice of the Offspring Population Size in Evolutionary Algorithms on the Choice of the Offspring Population Size in Evolutionary Algorithms , 2004 .

[10]  Frank Neumann,et al.  Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem , 2004, Eur. J. Oper. Res..

[11]  Carsten Witt,et al.  Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models , 2007, Evolutionary Computation.

[12]  Sven Oliver Krumke,et al.  On the Minimum Label Spanning Tree Problem , 1998, Inf. Process. Lett..

[13]  Frank Neumann,et al.  A Parameterized Runtime Analysis of Evolutionary Algorithms for the Euclidean Traveling Salesperson Problem , 2012, AAAI.

[14]  Yang Yu,et al.  An analysis on recombination in multi-objective evolutionary optimization , 2013, Artif. Intell..

[15]  Bruce L. Golden,et al.  A one-parameter genetic algorithm for the minimum labeling spanning tree problem , 2005, IEEE Transactions on Evolutionary Computation.

[16]  Yuren Zhou,et al.  A comparative runtime analysis of heuristic algorithms for satisfiability problems , 2009, Artif. Intell..

[17]  Frank Neumann,et al.  Approximating Minimum Multicuts by Evolutionary Multi-objective Algorithms , 2008, PPSN.

[18]  Frank Neumann,et al.  Computing Minimum Cuts by Randomized Search Heuristics , 2008, GECCO '08.

[19]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[20]  Xin Yao,et al.  A Large Population Size Can Be Unhelpful in Evolutionary Algorithms a Large Population Size Can Be Unhelpful in Evolutionary Algorithms , 2022 .

[21]  Gerhard J. Woeginger,et al.  Local search for the minimum label spanning tree problem with bounded color classes , 2003, Oper. Res. Lett..

[22]  Thomas Jansen,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods Evolutionary Algorithms-How to Cope With Plateaus of Constant Fitness and When to Reject Strings of the Same Fitness , 2001 .

[23]  Frank Neumann,et al.  Approximating Covering Problems by Randomized Search Heuristics Using Multi-Objective Models , 2010, Evolutionary Computation.

[24]  Benjamin Doerr,et al.  Crossover can provably be useful in evolutionary computation , 2008, GECCO '08.

[25]  Günther R. Raidl,et al.  Solving a k-Node Minimum Label Spanning Arborescence Problem to Compress Fingerprint Templates , 2009, J. Math. Model. Algorithms.

[26]  Bruce L. Golden,et al.  Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem , 2005, Oper. Res. Lett..

[27]  Pietro Simone Oliveto,et al.  Analysis of the $(1+1)$-EA for Finding Approximate Solutions to Vertex Cover Problems , 2009, IEEE Transactions on Evolutionary Computation.

[28]  Ingo Wegener,et al.  Real royal road functions--where crossover provably is essential , 2001, Discret. Appl. Math..

[29]  Frank Neumann,et al.  Minimum spanning trees made easier via multi-objective optimization , 2005, GECCO '05.

[30]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[31]  Bryant A. Julstrom,et al.  An effective genetic algorithm for the minimum-label spanning tree problem , 2006, GECCO '06.

[32]  Marco Laumanns,et al.  Running time analysis of multiobjective evolutionary algorithms on pseudo-Boolean functions , 2004, IEEE Transactions on Evolutionary Computation.

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

[34]  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 .

[35]  Oliver Giel,et al.  Expected runtimes of a simple multi-objective evolutionary algorithm , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[36]  Benjamin Doerr,et al.  Crossover can provably be useful in evolutionary computation , 2012, Theor. Comput. Sci..

[37]  Francisco Herrera,et al.  Tackling Real-Coded Genetic Algorithms: Operators and Tools for Behavioural Analysis , 1998, Artificial Intelligence Review.

[38]  Ruay-Shiung Chang,et al.  The Minimum Labeling Spanning Trees , 1997, Inf. Process. Lett..

[39]  Frank Neumann,et al.  Expected runtimes of evolutionary algorithms for the Eulerian cycle problem , 2004, Proceedings of the 2004 Congress on Evolutionary Computation (IEEE Cat. No.04TH8753).

[40]  Nenad Mladenovic,et al.  Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem , 2009, Eur. J. Oper. Res..

[41]  Xin Yao,et al.  Towards an analytic framework for analysing the computation time of evolutionary algorithms , 2003, Artif. Intell..

[42]  Stefan Voß,et al.  Metaheuristics Comparison for the Minimum Labelling Spanning Tree Problem , 2005 .

[43]  Malcolm Sambridge,et al.  Genetic algorithms: a powerful tool for large-scale nonlinear optimization problems , 1994 .

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

[45]  Xin Yao,et al.  Drift analysis and average time complexity of evolutionary algorithms , 2001, Artif. Intell..

[46]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .