Clean Citation Style 002
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Krejca, Martin S.; Witt, Carsten Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax. Foundations of Genetic Algorithms (FOGA) 2017: 6579
The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of \(Omega (\mu \sqrt n + n \log n)\), where \(\mu\) is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general blackbox complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.

Friedrich, Tobias; Kötzing, Timo; Quinzan, Francesco; Sutton, Andrew Michael Resampling vs Recombination: a Statistical Run Time Estimation. Foundations of Genetic Algorithms (FOGA) 2017: 2535
Noise is pervasive in realworld optimization, but there is still little understanding of the interplay between the operators of randomized search heuristics and explicit noisehandling techniques, such as statistical resampling. In this paper, we report on several statistical models and theoretical results that help to clarify this reciprocal relationship for a collection of randomized search heuristics on noisy functions. We consider the optimization of pseudoBoolean functions under additive posterior Gaussian noise and explore the tradeo between noise reduction and the computational cost of resampling. We first perform experiments to find the optimal parameters at a given noise intensity for a mutationonly evolutionary algorithm, a genetic algorithm employing recombination, an estimation of distribution algorithm (EDA), and an ant colony optimization algorithm. We then observe how the optimal parameter depends on the noise intensity for the different algorithms. Finally, we locate the point where statistical resampling costs more than it is worth in terms of run time. We find that the EA requires the highest number of resamples to obtain the best speedup, whereas crossover reduces both the run time and the number of resamples required. Most surprisingly, we find that EDAlike algorithms require no resampling, and can handle noise implicitly.

Friedrich, Tobias; Kötzing, Timo; Lagodzinski, J. A. Gregor; Neumann, Frank; Schirneck, Martin Analysis of the (1+1) EA on Subclasses of Linear Functions under Uniform and Linear Constraints. Foundations of Genetic Algorithms (FOGA) 2017: 4554
Linear functions have gained a lot of attention in the area of run time analysis of evolutionary computation methods and the corresponding analyses have provided many effective tools for analyzing more complex problems. In this paper, we consider the behavior of the classical (1+1) Evolutionary Algorithm for linear functions under linear constraint. We show tight bounds in the case where both the objective function and the constraint is given by the OneMax function and present upper bounds as well as lower bounds for the general case. Furthermore, we also consider the LeadingOnes fitness function.

Pourhassan, Mojgan; Friedrich, Tobias; Neumann, Frank On the Use of the Dual Formulation for Minimum Weighted Vertex Cover in Evolutionary Algorithms. Foundations of Genetic Algorithms (FOGA) 2017: 3744
We consider the weighted minimum vertex cover problem and investigate how its dual formulation can be exploited to design evolutionary algorithms that provably obtain a 2approximation. Investigating multivalued representations, we show that variants of randomized local search and the (1+1) EA achieve this goal in expected pseudopolynomial time. In order to speed up the process, we consider the use of step size adaptation in both algorithms and show that RLS obtains a 2approximation in expected polynomial time while the (1+1) EA still encounters a pseudopolynomial lower bound.

Sutton, Andrew M.; Whitley, Darrell; Howe, Adele E. Approximating the distribution of fitness over hamming regions. Foundations of Genetic Algorithms (FOGA) 2011: 93104
The distribution of fitness values across a set of states sharply influences the dynamics of evolutionary processes and heuristic search in combinatorial optimization. In this paper we present a method for approximating the distribution of fitness values over Hamming regions by solving a linear programming problem that incorporates low order moments of the target function. These moments can be retrieved in polynomial time for select problems such as MAX\(k\)SAT using Walsh analysis. The method is applicable to any real function on binary strings that is epistatically bounded and discrete with asymptotic bounds on the cardinality of its codomain. We perform several studies on the ONEMAX and MAX\(k\)SAT domains to assess the accuracy of the approximation and its dependence on various factors. We show that the approximation can accurately predict the number of states within a Hamming region that have an improving fitness value.

Kötzing, Timo; Neumann, Frank; Sudholt, Dirk; Wagner, Markus Simple maxmin ant systems and the optimization of linear pseudoboolean functions. Foundations of Genetic Algorithms (FOGA) 2011: 209218
With this paper, we contribute to the understanding of ant colony optimization (ACO) algorithms by formally analyzing their runtime behavior. We study simple MAXMIN ant systems on the class of linear pseudoBoolean 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 \(O((n^3 \log n)\rho)\) on the running time for two ACO variants on all linear functions, where \(\rho\) determines the pheromone update strength. Furthermore, we show improved bounds for two wellknown linear pseudoBoolean functions called ONEMAX and BINVAL and give additional insights using an experimental study.

Baswana, Surender; Biswas, Somenath; Doerr, Benjamin; Friedrich, Tobias; Kurur, Piyush P.; Neumann, Frank Computing Single Source Shortest Paths using SingleObjective Fitness Functions. Foundations of Genetic Algorithms (FOGA) 2009: 5966
Runtime analysis of evolutionary algorithms has become an important part in the theoretical analysis of randomized search heuristics. The first combinatorial problem where rigorous runtime results have been achieved is the wellknown single source shortest path (SSSP) problem. Scharnow, Tinnefeld and Wegener [PPSN 2002, J. Math. Model. Alg. 2004] proposed a multiobjective approach which solves the problem in expected polynomial time. They also suggest a related singleobjective fitness function. However, it was left open whether this does solve the problem efficiently, and, in a broader context, whether multiobjective fitness functions for problems like the SSSP yield more efficient evolutionary algorithms. In this paper, we show that the single objective approach yields an efficient (1+1) EA with runtime bounds very close to those of the multiobjective approach.

Bringmann, Karl; Friedrich, Tobias Don't be greedy when calculating hypervolume contributions. Foundations of Genetic Algorithms (FOGA) 2009: 103112
Most hypervolume indicator based optimization algorithms like SIBEA [Zitzler et al. 2007], SMSEMOA [Beume et al. 2007], or MOCMAES [Igel et al. 2007] remove the solution with the smallest loss with respect to the dominated hypervolume from the population. This is usually iterated \(\lambda\) times until the size of the population no longer exceeds a fixed size \(\mu\). We show that there are populations such that the contributing hypervolume of the \(\lambda\) solutions chosen by this greedy selection scheme can be much smaller than the contributing hypervolume of an optimal set of \(\lambda\) solutions. Selecting the optimal \(\lambda\)set implies calculating \(\frac{\mu + \lambda}{\mu}\) conventional hypervolume contributions, which is considered computationally too expensive. We present the first hypervolume algorithm which calculates directly the contribution of every set of \(\lambda\) solutions. This gives an additive term of \(\frac{\mu + \lambda}{\mu}\) in the runtime of the calculation instead of a multiplicative factor of binomial \(\frac{\mu + \lambda}{\mu}\) . Given a population of size\(n = \mu + \lambda \) our algorithm can calculate a set of \(\lambda \geq 1 \) solutions with minimal ddimensional hypervolume contribution in time \(O(n^{d/2 \log n + n^\lambda)\) for \(d > 2\). This improves all previously published algorithms by a factor of order \(n^{\min(\lambda, d/2)}\) for \(d > 3\). Therefore even if we remove the solutions one by one greedily as usual, we gain a speedup factor of \(n\) for all \(d > 3\).