Computational complexity analysis of genetic programming

Genetic Programming (GP) is an evolutionary computation technique to solve problems in an automated, domain-independent way. Rather than identifying the optimum of a function as in more traditional evolutionary optimization, the aim of GP is to evolve computer programs with a given functionality. A population of programs is evolved using variation operators inspired by Darwinian evolution (crossover and mutation) and natural selection principles to guide the search process towards better programs. While many GP applications have produced human competitive results, the theoretical understanding of what problem characteristics and algorithm properties allow GP to be effective is comparatively limited. Compared to traditional evolutionary algorithms for function optimization, GP applications are further complicated by two additional factors: the variable length representation of candidate programs, and the difficulty of evaluating their quality efficiently. Such difficulties considerably impact the runtime analysis of GP where space complexity also comes into play. As a result initial complexity analyses of GP focused on restricted settings such as evolving trees with given structures or estimating the quality of solutions using only a small polynomial number of input/output examples. However, the first runtime analyses concerning GP applications for evolving proper functions with defined input/output behavior have recently appeared. In this chapter, we present an overview of the state-of-the-art.