On Easiest Functions for Somatic Contiguous Hypermutations And Standard Bit Mutations

Understanding which function classes are easy and which are hard for a given algorithm is a fundamental question for the analysis and design of bio-inspired search heuristics. A natural starting point is to consider the easiest and hardest functions for an algorithm. For the (1+1)EA using standard bit mutation it is well known that OneMax is an easiest function with unique optimum while Trap is a hardest. In this paper we extend the analysis of easiest function classes to the contiguous somatic hypermutation (CHM) operator used in artificial immune systems. We define a function MinBlocks and prove that it is an easiest function for the (1+1)EA using CHM, presenting both a runtime and a fixed budget analysis. Since MinBlocks is, up to a factor of 2, a hardest function for standard bit mutations, we consider the effects of combining both operators into a hybrid algorithm. We show that an easiest function for the hybrid algorithm is not just a trivial weighted combination of the respective easiest functions for each operator. Nevertheless, by combining the advantages of both operators, the hybrid algorithm has optimal asymptotic performance on both OneMax and MinBlocks.

[1]  Thomas Jansen,et al.  Design and Management of Complex Technical Processes and Systems by means of Computational Intelligence Methods On the Choice of the Mutation Probability for the ( 1 + 1 ) EA , 2006 .

[2]  Benjamin Doerr,et al.  Drift analysis and linear functions revisited , 2010, IEEE Congress on Evolutionary Computation.

[3]  Xin-She Yang,et al.  Introduction to Algorithms , 2021, Nature-Inspired Optimization Algorithms.

[4]  Thomas Jansen,et al.  Analyzing different variants of immune inspired somatic contiguous hypermutations , 2011, Theor. Comput. Sci..

[5]  Thomas Jansen,et al.  Performance analysis of randomised search heuristics operating with a fixed budget , 2014, Theor. Comput. Sci..

[6]  Jonathan Timmis,et al.  Artificial Immune Systems: A New Computational Intelligence Approach , 2003 .

[7]  Thomas Jansen,et al.  Computing Longest Common Subsequences with the B-Cell Algorithm , 2012, ICARIS.

[8]  Per Kristian Lehre,et al.  Runtime analysis of selection hyper-heuristics with classical learning mechanisms , 2014, 2014 IEEE Congress on Evolutionary Computation (CEC).

[9]  Thomas Jansen,et al.  Analyzing Evolutionary Algorithms: The Computer Science Perspective , 2012 .

[10]  Dirk Sudholt,et al.  Analysis of an Iterated Local Search Algorithm for Vertex Coloring , 2010, ISAAC.

[11]  Jonathan Timmis,et al.  Immune Inspired Somatic Contiguous Hypermutation for Function Optimisation , 2003, GECCO.

[12]  Jonathan Timmis,et al.  Artificial immune systems - a new computational intelligence paradigm , 2002 .

[13]  Thomas Jansen,et al.  Evolutionary algorithms and artificial immune systems on a bi-stable dynamic optimisation problem , 2014, GECCO.

[14]  Thomas Jansen,et al.  On the Analysis of the Immune-Inspired B-Cell Algorithm for the Vertex Cover Problem , 2011, ICARIS.

[15]  Xin Yao,et al.  On the Easiest and Hardest Fitness Functions , 2012, IEEE Transactions on Evolutionary Computation.

[16]  John R. Woodward,et al.  Hyper-Heuristics , 2015, GECCO.

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

[18]  Thomas Jansen,et al.  Reevaluating Immune-Inspired Hypermutations Using the Fixed Budget Perspective , 2014, IEEE Transactions on Evolutionary Computation.

[19]  Dirk Sudholt,et al.  Hybridizing Evolutionary Algorithms with Variable-Depth Search to Overcome Local Optima , 2011, Algorithmica.

[20]  Thomas Jansen,et al.  Analyzing Evolutionary Algorithms , 2015, Natural Computing Series.

[21]  Dirk Sudholt,et al.  The impact of parametrization in memetic evolutionary algorithms , 2009, Theor. Comput. Sci..

[22]  Johannes Lengler,et al.  Fixed Budget Performance of the (1+1) EA on Linear Functions , 2015, FOGA.

[23]  Per Kristian Lehre,et al.  A runtime analysis of simple hyper-heuristics: to mix or not to mix operators , 2013, FOGA XII '13.

[24]  Pablo Moscato,et al.  Handbook of Memetic Algorithms , 2011, Studies in Computational Intelligence.