Nils Hebbinghaus

Do additional objectives make a problem harder

In this paper, we examine how adding objectives to a given optimization problem affects the computation effort required to generate the set of Pareto-optimal solutions. Experimental studies show that additional objectives may change the runtime behavior of an algorithm drastically. Often it is assumed that more objectives make a problem harder as the number of different trade-offs may increase with the problem dimension. We show that additional objectives, however, may be both beneficial and obstructive depending on the chosen objective. Our results are obtained by rigorous runtime analyses that show the different effects of adding objectives to a well-known plateau-function.

Approximating covering problems by randomized search heuristics using multi-objective models

The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical ones on this subject. We consider the approximation ability of randomized search heuristics for the class of covering problems and compare single-objective and multi-objective models for such problems. For the Vertex-Cover problem, we point out situations where the multi-objective model leads to a fast construction of optimal solutions while in the single-objective case even no good approximation can be achieved within expected polynomial time. Examining the more general Set-Cover problem we show that optimal solutions can be approximated within a factor of log n, where n is the problem dimension, using the multi-objective approach while the approximation quality obtainable by the single-objective approach in expected polynomial time may be arbitrarily bad.

On the Effects of Adding Objectives to Plateau Functions

In this paper, we examine how adding objectives to a given optimization problem affects the computational effort required to generate the set of Pareto-optimal solutions. Experimental studies show that additional objectives may change the running time behavior of an algorithm drastically. Often it is assumed that more objectives make a problem harder as the number of different tradeoffs may increase with the problem dimension. We show that additional objectives, however, may be both beneficial and obstructive depending on the chosen objective. Our results are obtained by rigorous running time analyses that show the different effects of adding objectives to a well-known plateau function. Additional experiments show that the theoretically shown behavior can be observed for problems with more than one objective.

Approximating covering problems by randomized search heuristics using multi-objective models*

The main aim of randomized search heuristics is to produce good approximations of optimal solutions within a small amount of time. In contrast to numerous experimental results, there are only a few theoretical explorations on this subject. We consider the approximation ability of randomized search heuristics for the class of covering problems and compare single-objective and multi-objective models for such problems. For the VertexCover problem, we point out situations where the multi-objective model leads to a fast construction of optimal solutions while in the single-objective case, no good approximation can be achieved within the expected polynomial time. Examining the more general SetCover problem, we show that optimal solutions can be approximated within a logarithmic factor of the size of the ground set, using the multi-objective approach, while the approximation quality obtainable by the single-objective approach in expected polynomial time may be arbitrarily bad.

Speeding up evolutionary algorithms through asymmetric mutation operators

Successful applications of evolutionary algorithms show that certain variation operators can lead to good solutions much faster than other ones. We examine this behavior observed in practice from a theoretical point of view and investigate the effect of an asymmetric mutation operator in evolutionary algorithms with respect to the runtime behavior. Considering the Eulerian cycle problem we present runtime bounds for evolutionary algorithms using an asymmetric operator which are much smaller than the best upper bounds for a more general one. In our analysis it turns out that a plateau which both algorithms have to cope with changes its structure in a way that allows the algorithm to obtain an improvement much faster. In addition, we present a lower bound for the general case which shows that the asymmetric operator speeds up computation by at least a linear factor.

Analyses of simple hybrid algorithms for the vertex cover problem

Hybrid methods are very popular for solving problems from combinatorial optimization. In contrast, the theoretical understanding of the interplay of different optimization methods is rare. In this paper, we make a first step into the rigorous analysis of such combinations for combinatorial optimization problems. The subject of our analyses is the vertex cover problem for which several approximation algorithms have been proposed. We point out specific instances where solutions can (or cannot) be improved by the search process of a simple evolutionary algorithm in expected polynomial time.

Speeding up evolutionary algorithms through restricted mutation operators

We investigate the effect of restricting the mutation operator in evolutionary algorithms with respect to the runtime behavior. For the Eulerian cycle problem; we present runtime bounds on evolutionary algorithms with a restricted operator that are much smaller than the best upper bounds for the general case. It turns out that a plateau that both algorithms have to cope with is left faster by the new algorithm. In addition, we present a lower bound for the unrestricted algorithm which shows that the restricted operator speeds up computation by at least a linear factor.

Plateaus can be harder in multi-objective optimization

In recent years a lot of progress has been made in understanding the behavior of evolutionary computation methods for single- and multi-objective problems. Our aim is to analyze the diversity mechanisms that are implicitly used in evolutionary algorithms for multi-objective problems by rigorous runtime analyses. We show that, even if the population size is small, the runtime can be exponential where corresponding single-objective problems are optimized within polynomial time. To illustrate this behavior we analyze a simple plateau function in a first step and extend our result to a class of instances of the well-known SETCOVER problem.

A rigorous view on neutrality

Motivated by neutrality observed in natural evolution often redundant encodings are used in evolutionary algorithms. Many experimental studies have been carried out on this topic. In this paper we present a first rigorous runtime analysis on the effect of using neutrality. We consider a simple model where a layer of constant fitness is distributed in the search space and point out situations where the use of neutrality significantly influence the runtime of an evolutionary algorithm.

On improving approximate solutions by evolutionary algorithms

Hybrid methods are very popular for solving problems from combinatorial optimization. In contrast to this the theoretical understanding of the interplay of different optimization methods is rare. The aim of this paper is to make a first step into the rigorous analysis of such combinations for combinatorial optimization problems. The subject of our analyses is the vertex cover problem for which several approximation algorithms have been proposed. We point out specific instances where solutions can (or cannot) be improved by the search process of a simple evolutionary algorithm in expected polynomial time.