Benefits of Sexual Reproduction in Evolutionary Computation

Abstract

Evolutionary algorithms have proven to be highly successful for real world combinatorial optimization problems too complex to be handled by traditional algorithmic approaches. In stark contrast to their practical relevance, we only start to theoretically understand the stochastic processes which govern these optimization algorithms. The development of evolutionary algorithms was inspired by biological observations. In nature there seems to be an inherent advantage of sexual recombination compared to mere asexual reproduction. Thus many successful evolutionary algorithms in practice make use of recombination strategies. However, our theoretical understanding of crossover in evolutionary computation is very limited. This thesis studies the behavior of several evolutionary algorithms with a focus on the benefits of sexual reproduction. Two archetypical combinatorial optimization problems are the NP-hard Traveling Salesperson Problem (TSP) and the polynomial-time solvable all-pairs shortest path problem (APSP). Both are representatives of the large class of combinatorial optimization problems with a dynamic programming structure. Chapter 2 of this thesis presents a mathematical description of the common properties of this class and derives a generic representation. This enables a prototypical evolutionary algorithm (without crossover) to construct solutions in a dynamic programming fashion without access to the structure of the problem. We prove upper bounds on the running time of this algorithm. This is the second time ever that runtime bounds of evolutionary algorithms for a large class of problems could be proven. Empirical experiments on TSP show that the use of crossover can lead to additional speed-ups. Genetic algorithms are evolutionary algorithms which employ sexual reproduction in form of some crossover operators. In order to get a better understanding of the working principles of crossover, Chapter 3 analyzes the runtime of genetic algorithms on common artificial pseudo-Boolean functions. Theoretical and empirical results show substantial speedups for functions of unitation when combined with a fitness-invariant bit shuffling operator. We also consider a simple genetic algorithm without shuffling and investigate the interplay of mutation and crossover on the test function Jump. We prove for small crossover probabilities that subsequent mutations create sufficient diversity, even for very small populations. Contrarily, with high crossover probabilities crossover tends to lose diversity more quickly than mutation can create it. Both effects have a drastic impact on the performance on Jump. We complement our theoretical findings by Monte Carlo simulations.