Optimizing an Arbitrary Function is Hard for the Genetic Algorithm

The Genetic Algorithm (GA) is generally portrayed as a search procedure which can optimize pseudo-boolean functions based on a limited sample of the function's values. There have been many attempts to analyze the computational behavior of the GA. For the most part, these attempts have tacitly assumed that the algorithmic parameters of the GA (e.g. population size, choice of genetic operators, etc.) can be isolated from the characteristics of the class of functions being optimized. In the following, we demonstrate why this assumption is inappropriate. We consider the class, F, of all deterministic pseudo-boolean functions whose values range over the integers. We then consider the Genetic Algorithm as a combinatorial optimization problem over f0; 1g l and demonstrate that the computational problem it attempts to solve is NP-hard relative to this class of functions. Using standard performance measures, we also give evidence that the Genetic Algorithm will not be able to eeciently approximate this optimization problem. These results imply that there does not exist a xed set of algorithmic parameters which enable the GA to optimize an arbitrary function in F. We conclude that theoretical and experimental analyses of the GA which do not specify the class of functions being optimized can make few claims regarding the eeciency of the genetic algorithm for an arbitrary tness function. When analyzing the computational complexity of the Genetic Algorithm, classes (or distributions) of functions should be analyzed relative to the algorithmic parameters chosen for the GA.