From TwoMax to the Ising Model: Easy and Hard Symmetrical Problems Clarissa Van Hoyweghen Intelligent Systems Lab University of Antwerp Groenenborgerlaan 171 2020 Antwerp, Belgium hoyweghe@ruca.ua.ac.be David E. Goldberg Illinois Genetic Algorithms Laboratory University of Illinois 117 Transportation Building 104 S. Mathews Av. Urbana, IL 61801 deg@illigal.ge.uiuc.edu Bart Naudts Intelligent Systems Lab University of Antwerp Groenenborgerlaan 171 2020 Antwerp, Belgium bnaudts@ruca.ua.ac.be Abstract The paper shows that there is a key dividing line between two types of symmetrical problems: problems for which a genetic algorithm (GA) benefits from the fact that genetic drift chooses between equally good partial solutions, and prob- lems for which all equally good partial solutions have to be preserved to find an optimum. By an- alyzing in detail the search process of a selec- torecombinative GA optimizing a TwoMax and comparing this search process with that of a one- dimensional Ising model, the paper investigates the difference between these two types of sym- metrical problems. For the first type of problems, naively adding a niching technique to the genetic algorithm makes the problem harder to solve. For the last type of problems, niching is necessary to find an optimum. 1 INTRODUCTION In the context of optimization by genetic algorithms (GAs), scaling, deception, epistasis, and noise are well known ex- amples of problem difficulty characteristics. The presence of one such characteristic in the representation of a search problem indicates a certain type of difficulty the GA is to encounter during its search for global optima. In this paper we investigate another aspect of problem difficulty: the ex- istence of equally good partial solutions in symmetrical or hierarchical problems. The loss of some equally good par- tial solutions due to genetic drift can prevent a GA to find an optimum, but for some problems it can also be benefi- cial. The purpose of this paper is twofold. Firstly, the paper an- alyzes the search process of a GA solving a multimodal equivalent of the OneMax problem, the so called TwoMax, and compares its search with the search process of a GA solving a OneMax. Secondly, the paper shows that there is a key dividing line between two different types of symmet- rical problems: problems for which the GA benefits from the fact that genetic drift chooses between equally good partial solutions, and problems for which all equally good partial solutions have to be preserved to find an optimum and for which niching becomes a necessity. Naively adding a niching technique to an algorithm optimizing a problem of the first type makes the problem harder to solve. The paper is structured as follows. Section 2 introduces equally good partial solutions or non-inferior BBs and shows how genetic drift chooses between them. Section 3 analyzes the search process of an easy symmetrical prob- lem, a TwoMax, and shows that there exists a class of prob- lems that benefits from the fact that genetic drift chooses between equally good partial solutions. Section 4 and 5 compare two types of symmetrical problems: problems for which searching in terms of non-inferior BBs is necessary and problems that benefit from the fact that genetic drift chooses between non-inferior BBs. Section 6 summarizes and concludes the paper. 2 NON-INFERIOR BBS AND NICHING The working of a genetic algorithm can be explained by the search for superior building blocks. Building blocks (BBs) with above average fitness are combined to con- struct higher order building blocks. However, recent stud- ies [16, 12, 7, 14] show that the search for superior BBs is not always sufficient to find a solution. When an op- timization problem contains BBs which are equally good and superior to all alternatives, so called non-inferior BBs [4], making a choice between such BBs can avoid a GA to reach an optimum. A search strategy preserving all non- inferior BBs can then become a necessity to solve the prob- lem quickly, reliably, and accurately. In [14], for exam- ple, it is shown that the simple GA cannot solve the stan- dard Ising model in a reasonable amount of time because the population loses some non-inferior BBs due to genetic