Performance Analysis of the (1+1) Evolutionary Algorithm for the Multiprocessor Scheduling Problem

In recent years, there has been considerable progress in the theoretical study of evolutionary algorithms (EAs) for discrete optimization problems. However, results on the performance analysis of EAs for NP-hard problems are rare. This paper contributes a theoretical understanding of EAs on the NP-hard multiprocessor scheduling problem. The worst-case bound on the (1+1)EA for the multiprocessor scheduling problem and a worst-case example are presented. It is proved that the (1+1)EA on $$Q2\mid \mid C_\mathrm{max}$$Q2∣∣Cmax problem achieves an approximation ratio of $$\frac{1+\sqrt{5}}{2}$$1+52 in expected time $$O(n^2)$$O(n2). Finally, the theoretical analysis on three selected instances of the multiprocessor scheduling problem shows that EAs outperform local search algorithms on these instances.

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

[2]  Kathryn E. Stecke,et al.  Design, planning, scheduling, and control problems of flexible manufacturing systems , 1985 .

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

[4]  Benjamin Doerr,et al.  Tight Analysis of the (1+1)-EA for the Single Source Shortest Path Problem , 2011, Evolutionary Computation.

[5]  E.L. Lawler,et al.  Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey , 1977 .

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

[7]  Oscar H. Ibarra,et al.  Bounds for LPT Schedules on Uniform Processors , 1977, SIAM J. Comput..

[8]  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 .

[9]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[10]  Christian Gunia,et al.  On the analysis of the approximation capability of simple evolutionary algorithms for scheduling problems , 2005, GECCO '05.

[11]  Frank Neumann,et al.  Analyses of Simple Hybrid Algorithms for the Vertex Cover Problem , 2009, Evolutionary Computation.

[12]  Benjamin Doerr,et al.  A tight analysis of the (1 + 1)-EA for the single source shortest path problem , 2007, IEEE Congress on Evolutionary Computation.

[13]  David B. Shmoys,et al.  Using dual approximation algorithms for scheduling problems: Theoretical and practical results , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[14]  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 .

[15]  Xin Yao,et al.  From an individual to a population: an analysis of the first hitting time of population-based evolutionary algorithms , 2002, IEEE Trans. Evol. Comput..

[16]  Xin Yao,et al.  Time complexity of evolutionary algorithms for combinatorial optimization: A decade of results , 2007, Int. J. Autom. Comput..

[17]  Thomas Bäck,et al.  Evolutionary algorithms in theory and practice - evolution strategies, evolutionary programming, genetic algorithms , 1996 .

[18]  Frank Neumann,et al.  Bioinspired computation in combinatorial optimization: algorithms and their computational complexity , 2010, GECCO '12.

[19]  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).

[20]  Ellis Horowitz,et al.  A linear time approximation algorithm for multiprocessor scheduling , 1979 .

[21]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[22]  Stephen R. Marsland,et al.  Convergence Properties of (μ + λ) Evolutionary Algorithms , 2011, AAAI.

[23]  Chris N. Potts,et al.  Fifty years of scheduling: a survey of milestones , 2009, J. Oper. Res. Soc..

[24]  Michael A. Langston,et al.  Improved 0/1-interchange scheduling , 1982, BIT.

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

[26]  Frank Neumann,et al.  Computing single source shortest paths using single-objective fitness , 2009, FOGA '09.

[27]  Sanjoy K. Baruah,et al.  A Categorization of Real-Time Multiprocessor Scheduling Problems and Algorithms , 2004, Handbook of Scheduling.

[28]  Petra Schuurman,et al.  Performance Guarantees of Local Search for Multiprocessor Scheduling , 2001, INFORMS J. Comput..

[29]  Emile H. L. Aarts,et al.  Theoretical aspects of local search , 2006, Monographs in Theoretical Computer Science. An EATCS Series.

[30]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

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

[32]  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 .

[33]  Thomas Jansen,et al.  On the analysis of the (1+1) evolutionary algorithm , 2002, Theor. Comput. Sci..

[34]  Leon F. McGinnis,et al.  A large scale machine loading problem in flexible assembly , 1985 .

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

[36]  Emile H. L. Aarts,et al.  Theoretical Aspects of Local Search (Monographs in Theoretical Computer Science. An EATCS Series) , 2007 .

[37]  J. Kingman.  A FIRST COURSE IN STOCHASTIC PROCESSES , 1967 .

[38]  Sartaj Sahni,et al.  Bounds for List Schedules on Uniform Processors , 1980, SIAM J. Comput..

[39]  Frank Neumann,et al.  A Parameterized Runtime Analysis of Simple Evolutionary Algorithms for Makespan Scheduling , 2012, PPSN.