On Easiest Functions for Somatic Contiguous Hypermutations And Standard Bit Mutations

Abstract

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.

Theoretical Foundations of Immune-Inspired Randomized Search Heuristics for Optimization

“…Very recently, Corus et al [8,9] derived an easiest function for contiguous hypermutations. Again using the insight that contiguous hypermutations can have advantages on functions that require mutations of many bits simultaneously, they introduced the following fitness function.…”

“…The second best bit string is 0 n , with fitness l − 1. Corus et al [8,9] presented both runtime and fixed-budget analyses of contiguous hypermutations on this function and showed that MinBlocks is indeed an easiest function using a method introduced by He et al [28]. The runtime of contiguous hypermutations embedded in a (1+1) framework (see Algorithm 1) on MinBlocks is Θ(n 2 ).…”

“…Finally, Corus et al [8,9] discussed a number of hybridizations of standard bit mutations and contiguous hypermutations. These allow one to combine the advantages of the two operators, and yield optimal asymptotic performance on both OneMax and MinBlocks.…”

Expected Fitness Gains of Randomized Search Heuristics for the Traveling Salesperson Problem

“…Given such a strong relation between fitness and time, and the availability of upper and lower tail bounds, it is then possible to derive tight upper and lower bounds on the expected fitness increase over any given period of time. More recent studies in this line of research include comparisons of fixed budget results obtained by evolutionary algorithms to other RSH such as artificial immune systems [5,16], fixed budget analysis in dynamic optimization [14] and introduction of more general methods of analysis [18].…”

On Easiest Functions for Mutation Operators in Bio-Inspired Optimisation

“…In Sect. 4 we present an analysis of the optimisation time as well as a fixed budget analysis for CHM on MinBlocks. We then show that SBM alone is not able to optimise the constructed function and that a hybridisation of SBM and CHM can have significant advantages (Sect.…”