Non-Elitist Genetic Algorithm as a Local Search Method

The genetic algorithm (GA) proposed by J. Holland [10] is a randomized heuristic search method, based on analogy with the genetic mechanisms observed in nature and employing a population of tentative solutions. Different modifications of GA are widely used in the areas of operations research pattern recognition, artificial intelligence etc. (see e.g. [13, 16]). Despite of numerous experimental investigations of these algorithms, their theoretical analysis is still at an early stage [5]. In this paper, the genetic algorithms are studied from the prospective of local search for combinatorial optimization problems, and the NP optimization problems in particular [2]. The major attention is payed to identification of the situations where the GA finds a local optimum in polynomially bounded time on average. Here and below we assume that the randomness is generated only by the randomized operators of selection, crossover and mutation within the GA. In what follows, we call a value polynomially bounded, if there exists a polynomial in the length of the problem input, which bounds the value from above. Throughout the paper we use the terms efficient algorithm or polynomial-time algorithm for an algorithm with polynomially bonded running time. A problem which is solved by such an algorithm is polynomially solvable. This study is motivated by the fact that the GAs are often considered to be the local search methods (see e.g. [1, 11, 14]). Therefore a topical question is: In what circumstances the GA efficiency is due to its similarity with the local search?