Timo Kötzing

How crossover helps in pseudo-boolean optimization

Understanding the impact of crossover on performance is a major problem in the theory of genetic algorithms (GAs). We present new insight on working principles of crossover by analyzing the performance of crossover-based GAs on the simple functions OneMax and Jump. First, we assess the potential speedup by crossover when combined with a fitness-invariant bit shuffling operator that simulates a lineage of independent evolution on a function of unitation. Theoretical and empirical results show drastic speedups for both functions. Second, we consider a simple GA without shuffling and investigate the interplay of mutation and crossover on Jump. If the crossover probability is small, 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. This has a drastic impact on the performance on Jump. We complement our theoretical findings by Monte Carlo simulations on the population diversity.

Faster black-box algorithms through higher arity operators

We extend the work of Lehre and Witt (GECCO 2010) on the unbiased black-box model by considering higher arity variation operators. In particular, we show that already for binary operators the black-box complexity of LeadingOnes drops from Θ(<i>n</i>²) for unary operators to <i>O</i>(<i>n</i> log <i>n</i>). For OneMax, the Ω(<i>n</i> log <i>n</i>) unary black-box complexity drops to <i>O</i>(<i>n</i>) in the binary case. For <i>k</i>-ary operators, <i>k</i> ≤ <i>n</i>, the OneMax-complexity further decreases to <i>O</i>(<i>n</i> / log <i>k</i>).

Theoretical Analysis of Two ACO Approaches for the Traveling Salesman Problem

Bioinspired algorithms, such as evolutionary algorithms and ant colony optimization, are widely used for different combinatorial optimization problems. These algorithms rely heavily on the use of randomness and are hard to understand from a theoretical point of view. This paper contributes to the theoretical analysis of ant colony optimization and studies this type of algorithm on one of the most prominent combinatorial optimization problems, namely the traveling salesperson problem (TSP). We present a new construction graph and show that it has a stronger local property than one commonly used for constructing solutions of the TSP. The rigorous runtime analysis for two ant colony optimization algorithms, based on these two construction procedures, shows that they lead to good approximation in expected polynomial time on random instances. Furthermore, we point out in which situations our algorithms get trapped in local optima and show where the use of the right amount of heuristic information is provably beneficial.

Ants easily solve stochastic shortest path problems

The first rigorous theoretical analysis (Horoba, Sudholt (GECCO 2010)) of an ant colony optimizer for the stochastic shortest path problem suggests that ant system experience significant difficulties when the input data is prone to noise. In this work, we propose a slightly different ant optimizer to deal with noise. We prove that under mild conditions, it finds the paths with shortest expected length efficiently, despite the fact that we do not have convergence in the classic sense. To prove our results, we introduce a stronger drift theorem that can also deal with the situation that the progress is faster when one is closer to the goal.

(1+1) EA on Generalized Dynamic OneMax

Evolutionary algorithms (EAs) perform well in settings involving uncertainty, including settings with stochastic or dynamic fitness functions. In this paper, we analyze the (1+1) EA on dynamically changing OneMax, as introduced by Droste (2003). We re-prove the known results on first hitting times using the modern tool of drift analysis. We extend these results to search spaces which allow for more than two values per dimension. Furthermore, we make an anytime analysis as suggested by Jansen and Zarges (2014), analyzing how closely the (1+1) EA can track the dynamically moving optimum over time. We get tight bounds both for the case of bit strings, as well as for the case of more than two values per position. Surprisingly, in the latter setting, the expected quality of the search point maintained by the (1+1) EA does not depend on the number of values per dimension.

ACO beats EA on a dynamic pseudo-boolean function

In this paper, we contribute to the understanding of the behavior of bio-inspired algorithms when tracking the optimum of a dynamically changing fitness function over time. In particular, we are interested in the difference between a simple evolutionary algorithm (EA) and a simple ant colony optimization (ACO) system on deterministically changing fitness functions, which we call dynamic fitness patterns. Of course, the algorithms have no prior knowledge about the patterns. We construct a bit string optimization problem where we can show that the ACO system is able to follow the optimum while the EA gets lost.

Optimizing expected path lengths with ant colony optimization using fitness proportional update

We study the behavior of a Max-Min Ant System (MMAS) on the stochastic single-destination shortest path (SDSP) problem. Two previous papers already analyzed this setting for two slightly different MMAS algorithms, where the pheromone update fitness-independently rewards edges of the best-so-far solution. The first paper showed that, when the best-so-far solution is not reevaluated and the stochastic nature of the edge weights is due to noise, the MMAS will find a tree of edges successfully and efficiently identify a shortest path tree with minimal noise-free weights. The second paper used reevaluation of the best-so-far solution and showed that the MMAS finds paths which beat any other path in direct comparisons, if existent. For both results, for some random variables, this corresponds to a tree with minimal expected weights. In this work we analyze a variant of MMAS that works with fitness-proportional update on stochastic-weight graphs with arbitrary random edge weights from [0,1]. For δ such that any suboptimal path is worse by at least δ than an optimal path, then, with suitable parameters, the graph will be optimized after O(n3 ln n/δ over δ3 iterations (in expectation). In order to prove the above result, the multiplicative and the variable drift theorem are adapted to continuous search spaces.

Simple max-min ant systems and the optimization of linear pseudo-boolean functions

With this paper, we contribute to the understanding of ant colony optimization (ACO) algorithms by formally analyzing their runtime behavior. We study simple MAX-MIN ant systems on the class of linear pseudo-Boolean functions defined on binary strings of length n. Our investigations point out how the progress according to function values is stored in the pheromones. We provide a general upper bound of O((n3 log n)ρ) on the running time for two ACO variants on all linear functions, where ρ determines the pheromone update strength. Furthermore, we show improved bounds for two well-known linear pseudo-Boolean functions called ONEMAX and BINVAL and give additional insights using an experimental study.

More effective crossover operators for the all-pairs shortest path problem

The all-pairs shortest path problem is the first non-artificial problem for which it was shown that adding crossover can significantly speed up a mutation-only evolutionary algorithm. Recently, the analysis of this algorithm was refined and it was shown to have an expected optimization time (w. r. t. the number of fitness evaluations) of @Q(n^3^.^2^5(logn)^0^.^2^5). In contrast to this simple algorithm, evolutionary algorithms used in practice usually employ refined recombination strategies in order to avoid the creation of infeasible offspring. We study extensions of the basic algorithm by two such concepts which are central in recombination, namely repair mechanisms and parent selection. We show that repairing infeasible offspring leads to an improved expected optimization time of O(n^3^.^2(logn)^0^.^2). As a second part of our study we prove that choosing parents that guarantee feasible offspring results in an optimization time of O(n^3logn). Both results show that already simple adjustments of the recombination operator can asymptotically improve the runtime of evolutionary algorithms.

Theoretical Properties of Two ACO Approaches for the Traveling Salesman Problem

Ant colony optimization (ACO) has been widely used for different combinatorial optimization problems. In this paper, we investigate ACO algorithms with respect to their runtime behavior for the traveling salesperson (TSP) problem. We present a new construction graph and show that it has a stronger local property than the given input graph which is often used for constructing solutions. Later on, we investigate ACO algorithms for both construction graphs on random instances and show that they achieve a good approximation in expected polynomial time.