Clean Citation Style 002
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Shi, Feng; Schirneck, Martin; Friedrich, Tobias; Kötzing, Timo; Neumann, Frank Erratum to: Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints. Algorithmica 2019
In the article "Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dy namic Uniform Constraints", we claimed a worstcase runtime of \(O(nD \log D)\) and \(O(nD)\) for the MultiObjective Evolutionary Algorithm and the MultiObjective \((\mu+(\lambda, \lambda))\) Genetic Algorithm, respectively, on linear profit functions under dynamic uniform constraint, where \(D = B − B^*\) denotes the difference between the original constraint bound \(B\) and the new one \(B^*\) . The technique used to prove these results contained an error. We correct this mistake and show a weaker bound of \(O(nD^2)\) for both algorithms instead.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Solving Problems with Unknown Solution Length at Almost No Extra Cost. Algorithmica 2019: 703748

Shi, Feng; Schirneck, Martin; Friedrich, Tobias; Kötzing, Timo; Neumann, Frank Reoptimization Time Analysis of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints. Algorithmica 2019: 828857
Rigorous runtime analysis is a major approach towards understanding evolutionary computing techniques, and in this area linear pseudoBoolean objective functions play a central role. Having an additional linear constraint is then equivalent to the NPhard Knapsack problem, certain classes thereof have been studied in recent works. In this article, we present a dynamic model of optimizing linear functions under uniform constraints. Starting from an optimal solution with respect to a given constraint bound, we investigate the runtimes that different evolutionary algorithms need to recompute an optimal solution when the constraint bound changes by a certain amount. The classical \((1+1)\) EA and several populationbased algorithms are designed for that purpose, and are shown to recompute efficiently. Furthermore, a variant of the \((1+(\lambda,\lambda))\) GA for the dynamic optimization problem is studied, whose performance is better when the change of the constraint bound is small.

Doerr, Benjamin; Fischbeck, Philipp; Frahnow, Clemens; Friedrich, Tobias; Kötzing, Timo; Schirneck, Martin Island Models Meet Rumor Spreading. Algorithmica 2019: 886915
Island models in evolutionary computation solve problems by a careful interplay of independently running evolutionary algorithms on the island and an exchange of good solutions between the islands. In this work, we conduct rigorous run time analyses for such island models trying to simultaneously obtain good run times and low communication effort. We improve the existing upper bounds for both measures (i) by improving the run time bounds via a careful analysis, (ii) by balancing individual computation and communication in a more appropriate manner, and (iii) by replacing the usual communicatewithall approach with randomized rumor spreading. In the latter, each island contacts a randomly chosen neighbor. This epidemic communication paradigm is known to lead to very fast and robust information dissemination in many applications. Our results concern island models running simple (1+1) evolutionary algorithms to optimize the classic test functions OneMax and LeadingOnes. We investigate binary trees, ddimensional tori, and complete graphs as communication topologies.

Trubenova, Barbora; Kötzing, Timo; Krejca, Martin S.; Lehre, Per Kristian Surfing on the seascape: Adaptation in a changing environment. Evolution: International Journal of Organic Evolution 2019: 13561374
The environment changes constantly at various time scales and, in order to survive, species need to keep adapting. Whether these species succeed in avoiding extinction is a major evolutionary question. Using a multilocus evolutionary model of a mutation‐limited population adapting under strong selection, we investigate the effects of the frequency of environmental fluctuations on adaptation. Our results rely on an “adaptive‐walk” approximation and use mathematical methods from evolutionary computation theory to investigate the interplay between fluctuation frequency, the similarity of environments, and the number of loci contributing to adaptation. First, we assume a linear additive fitness function, but later generalize our results to include several types of epistasis. We show that frequent environmental changes prevent populations from reaching a fitness peak, but they may also prevent the large fitness loss that occurs after a single environmental change. Thus, the population can survive, although not thrive, in a wide range of conditions. Furthermore, we show that in a frequently changing environment, the similarity of threats that a population faces affects the level of adaptation that it is able to achieve. We check and supplement our analytical results with simulations.

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. Theoretical Computer Science 2019
Linear functions have gained great attention in the run time analysis of evolutionary computation methods. The corresponding investigations have provided many effective tools for analyzing more complex problems. So far, the runtime analysis of evolutionary algorithms has mainly focused on unconstrained problems, but problems occurring in applications frequently involve constraints. Therefore, there is a strong need to extend the methods for analyzing unconstrained problems to a setting involving constraints. In this paper, we consider the behavior of the classical (1+1) evolutionary algorithm on linear functions under linear constraint. We show tight bounds in the case where the constraint is given by the OneMax function and the objective function is given by either the OneMax or the BinVal function. For the general case we present upper and lower bounds.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S. Unbiasedness of EstimationofDistribution Algorithms. Theoretical Computer Science 2019: 4659
In the context of blackbox optimization, blackbox complexity is used for understanding the inherent difficulty of a given optimization problem. Central to our understanding of natureinspired search heuristics in this context is the notion of unbiasedness. Specialized blackbox complexities have been developed in order to better understand the limitations of these heuristics – especially of (populationbased) evolutionary algorithms (EAs). In contrast to this, we focus on a model for algorithms explicitly maintaining a probability distribution over the search space: socalled estimationofdistribution algorithms (EDAs). We consider the recently introduced \(n\)Bernoulli\(\lambda\)EDA framework, which subsumes, for example, the commonly known EDAs PBIL, UMDA, \(\lambda\)MMAS\(_\textrm{IB}\), and cGA. We show that an \(n\)Bernoulli\(\lambda\)EDA is unbiased if and only if its probability distribution satisfies a certain invariance property under isometric automorphisms of \([0, 1]^n\). By restricting how an \(n\)Bernoulli\(\lambda\)EDA can perform an update, in a way common to many examples, we derive conciser characterizations, which are easy to verify. We demonstrate this by showing that our examples above are all unbiased.

Fokina, Ekaterina; Kötzing, Timo; San Mauro, Luca Limit Learning Equivalence Structures. Algorithmic Learning Theory (ALT) 2019: 120
While most research in Goldstyle learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pairs of elements of a structure are related and which are not. The aim of the learner is to find (an effective description of) the isomorphism type of the structure presented in the limit. In accordance with language learning we call this learning criterion InfExlearning (explanatory learning from informant). We start with a discussion and separations of different variants of this learning criterion, including learning from text (where the only information provided is which elements are related, and not which elements are not related) and finite learning (where the first actual conjecture of the learner has to be correct). This gives first intuitions and examples for what (classes of) structures are learnable and which are not. Our main contribution is a complete characterization of the learnable classes of structures in terms of a combinatorial property of the classes. This property allows us to derive a bound of \(\mathbf{0''}\) on the computational complexity required to learn uniformly enumerable classes of structures. Finally, we show how learning classes of structures relates to learning classes of languages by mapping learning tasks for structures to equivalent learning tasks for languages.

Doerr, Benjamin; Kötzing, Timo Multiplicative UpDrift. Genetic and Evolutionary Computation Conference (GECCO) 2019
Drift analysis aims at translating the expected progress of an evo lutionary algorithm (or more generally, a random process) into a probabilistic guarantee on its run time (hitting time). So far, drift arguments have been successfully employed in the rigorous analy sis of evolutionary algorithms, however, only for the situation that the progress is constant or becomes weaker when approaching the target. Motivated by questions like how fast fit individuals take over a population, we analyze random processes exhibiting a multiplica tive growth in expectation. We prove a drift theorem translating this expected progress into a hitting time. This drift theorem gives a sim ple and insightful proof of the levelbased theorem first proposed by Lehre (2011). Our version of this theorem has, for the first time, the bestpossible linear dependence on the growth parameter \(\delta\) (the previousbest was quadratic). This gives immediately stronger run time guarantees for a number of applications.

Peters, Jannik; Stephan, Daniel; Amon, Isabel; Gawendowicz, Hans; Lischeid, Julius; Salabarria, Julius; Umland, Jonas; Werner, Felix; Krejca, Martin S.; Rothenberger, Ralf; Kötzing, Timo; Friedrich, Tobias Mixed Integer Programming versus Evolutionary Computation for Optimizing a Hard RealWorld Staff Assignment Problem. International Conference on Automated Planning and Scheduling (ICAPS) 2019: 541554
Assigning staff to engagements according to hard constraints while optimizing several objectives is a task encountered by many companies on a regular basis. Simplified versions of such assignment problems are NPhard. Despite this, a typical approach to solving them consists of formulating them as mixed integer programming (MIP) problems and using a stateoftheart solver to get solutions that closely approximate the optimum. In this paper, we consider a complex realworld staff assignment problem encountered by the professional service company KPMG, with the goal of finding an algorithm that solves it faster and with a better solution than a commercial MIP solver. We follow the evolutionary algorithm (EA) metaheuristic and design a search heuristic which iteratively improves a solution using domainspecific mutation operators. Furthermore, we use a flow algorithm to optimally solve a subproblem, which tremendously reduces the search space for the EA. For our realworld instance of the assignment problem, given the same total time budget of \(100\) hours, a parallel EA approach finds a solution that is only \(1.7\)% away from an upper bound for the (unknown) optimum within under five hours, while the MIP solver Gurobi still has a gap of \(10.5\) %.

Kötzing, Timo; Sudholt, Dirk Preface to the Special Issue on Theory of Genetic and Evolutionary Computation. Algorithmica 2018: 15751578
Evolutionary algorithms (EAs) are randomized search heuristics that can be employed to solve complex optimization problems, including multimodal or highly constrained problems. EAs work by mimicking principles from natural evolution: maintaining a collection of possible solutions (a population) and iteratively creating variants of the individuals (the offspring) and then choosing a new set of individuals for the next iteration (selection). EAs are popular because they represent generalpurpose optimizers that can be easily applied to various problems, even in cases where little or no indepth knowledge about the problem is available. In order to guide practitioners devising new and effective algorithms, theoretical computer scientists employ methods from the field of randomized algorithms to analyze the working principles of EAs with mathematical rigor. Key questions concern the impact of parameter choices (such as, for example, the offspring size or the choice of variation operators) as well as foundational work on developing powerful analysis methods. The theory track of the annual ACM Genetic and Evolutionary Computation Conference (GECCO) is the first tier event for advances in this direction. In this special issue six selected papers from the 2016 edition of the GECCO theory track are collected, each one of them carefully revised and extended to meet the high quality standards of Algorithmica.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Static and SelfAdjusting Mutation Strengths for Multivalued Decision Variables. Algorithmica 2018: 17321768
The most common representation in evolutionary computation are bit strings. With very little theoretical work existing on how to use evolutionary algorithms for decision variables taking more than two values, we study the run time of simple evolutionary algorithms on some OneMaxlike functions defined over \(\Omega={0,1,\dots,r−1}^n\). We observe a crucial difference in how we extend the onebitflip and standardbit mutation operators to the multivalued domain. While it is natural to modify a random position of the string or select each position of the solution vector for modification independently with probability \(1/n\), there are various ways to then change such a position. If we change each selected position to a random value different from the original one, we obtain an expected run time of \(\Theta(n r \log n)\). If we change each selected position by \(+1\) or \(−1\) (random choice), the optimization time reduces to \(\Theta(n r + n \log n)\). If we use a random mutation strength \(i \in {0,1,\dots,r−1}\) with probability inversely proportional to \(i\) and change the selected position by \(+i\) or \(−i\) (random choice), then the optimization time becomes \(\Theta(n \log(r)(\log n + \log r))\), which is asymptotically faster than the previous if \(r = \omega(\log(n)\log\log(n))\). Interestingly, a better expected performance can be achieved with a selfadjusting mutation strength that is based on the success of previous iterations. For the mutation operator that modifies a randomly chosen position, we show that the selfadjusting mutation strength yields an expected optimization time of \(\Theta(n(\log n + \log r))\), which is best possible among all dynamic mutation strengths. In our proofs, we use a new multiplicative drift theorem for computing lower bounds, which is not restricted to processes that move only towards the target.

Aschenbach, Martin; Kötzing, Timo; Seidel, Karen Learning from Informants: Relations between Learning Success Criteria. ArXiv 2018: 37
Learning from positive and negative information, socalled informants, being one of the models for human and machine learning introduced by Gold [1967], is investigated. Particularly, naturally arising questions about this learning setting, originating in results on learning from solely positive information, are answered. By a carefully arranged argument learners can be assumed to only change their hypothesis in case it is inconsistent with the data (such a learning behavior is called conservative). The deduced main theorem states the relations between the most important delayable learning success criteria, being the ones not ruined by a delayed in time hypothesis output. Additionally, our investigations concerning the nondelayable requirement of consistent learning underpin the claim for delayability being the right structural property to gain a deeper understanding concerning the nature of learning success criteria. Moreover, we obtain an anomalous hierarchy when allowing for an increasing finite number of anomalies of the hypothesized language by the learner compared with the language to be learned. In contrast to the vacillatory hierarchy for learning from solely positive information, we observe a duality depending on whether infinitely many vacillations between different (almost) correct hypotheses are still considered a successful learning behavior.

Dang, DucCuong; Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Lehre, Per Kristian; Oliveto, Pietro S.; Sudholt, Dirk; Sutton, Andrew M. Escaping Local Optima Using Crossover with Emergent Diversity. IEEE Transactions on Evolutionary Computation 2018: 484497
Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the \((\mu+1)\) GA using uniform crossover on the fitness function \(Jump_k\). All previous results in this context only hold for unrealistically low crossover probability \(p_c=O(k/n)\), while we give analyses for the setting of constant \(p_c < 1\) in all but one case. Our bounds show a dependence on the problem size \(n\), the jump length \(k\), the population size \(\mu\), and the crossover probability \(p_c\). For the typical case of constant \(k > 2\) and constant \(p_c\), we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of \(\mu\): \(O(n^{k1})\) for duplicate elimination/minimisation, \(O(n^2 \log n)\) for maximising the convex hull, \(O(n \log n)\) for det. crowding (assuming \(p_c = k/n\)), \(O(n \log n)\) for maximising the Hamming distance, \(O(n \log n)\) for fitness sharing, \(O(n \log n)\) for the singlereceiver island model. This proves a sizeable advantage of all variants of the \((\mu+1)\) GA compared to the (1+1) EA, which requires \(\Theta(n^k)\). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.

Friedrich, Tobias; Kötzing, Timo; Quinzan, Francesco; Sutton, Andrew M. Improving the Run Time of the (1+1) Evolutionary Algorithm with Luby Sequences. Genetic and Evolutionary Computation Conference (GECCO) 2018: 301308
In the context of black box optimization, the most common way to handle deceptive attractors is to periodically restart the algorithm. In this paper, we explore the benefits of combining the simple \((1+1)\) Evolutionary Algorithm (EA) with the Luby Universal Strategy  the \((1+1)~EA_{\mathcal{U}}\), a metaheuristic that does not require parameter tuning. We first consider two artificial pseudoBoolean landscapes, on which the \((1+1)~EA\) exhibits exponential run time. We prove that the \((1+1)~EA_{\mathcal{U}}\) has polynomial run time on both instances. We then consider the Minimum Vertex Cover on two classes of graphs. Again, the \((1+1)~EA\) yields exponential run time on those instances, and the \((1+1)~EA_{\mathcal{U}}\) finds the global optimum in polynomial time. We conclude by studying the Makespan Scheduling. We consider an instance on which the \((1+1)~EA\) does not find a \((4/3\epsilon)\)approximation in polynomial time, and we show that the \((1+1)~EA_{\mathcal{U}}\) reaches a \((4/3\epsilon)\)approximation in polynomial time. We then prove that the \((1+1)~EA_{\mathcal{U}}\) serves as an Efficient Polynomialtime Approximation Scheme (EPTAS) for the Partition Problem, for a \((1+\epsilon)\)approximation with \(\epsilon > 4/n\).

Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes; Melnichenko, Anna Destructiveness of Lexicographic Parsimony Pressure and Alleviation by a Concatenation Crossover in Genetic Programming. Parallel Problem Solving From Nature (PPSN) 2018: 4254
For theoretical analyses there are two specifics distinguishing GP from many other areas of evolutionary computation. First, the variable size representations, in particular yielding a possible bloat (i.e. the growth of individuals with redundant parts). Second, the role and realization of crossover, which is particularly central in GP due to the treebased representation. Whereas some theoretical work on GP has studied the effects of bloat, crossover had a surprisingly little share in this work. We analyze a simple crossover operator in combination with local search,where a preference for small solutions minimizes bloat (lexicographic parsimony pressure); the resulting algorithm is denoted ConcatenationCrossover GP. For this purpose three variants of the wellstudied Majority test function with large plateaus are considered. We show that the Concatenation Crossover GP can efficiently optimize these test functions,while local search cannot be efficient for all three variants independent of employing bloat control.

Kötzing, Timo; Krejca, Martin S. FirstHitting Times for Finite State Spaces. Parallel Problem Solving From Nature (PPSN) 2018: 7991
One of the most important aspects of a randomized algorithm is bounding its expected run time on various problems. Formally speaking, this means bounding the expected firsthitting time of a random process. The two arguably most popular tools to do so are the fitness level method and drift theory. The fitness level method considers arbitrary transition probabilities but only allows the process to move toward the goal. On the other hand, drift theory allows the process to move into any direction as long as it moves closer to the goal in expectation; however, this tendency has to be monotone and, thus, the transition probabilities cannot be arbitrary. We provide a result that combines the benefit of these two approaches: our result gives a lower and an upper bound for the expected firsthitting time of a random process over \(\{0, \ldots ,n\}\) that is allowed to move forward and backward by 1 and can use arbitrary transition probabilities. In case that the transition probabilities are known, our bounds coincide and yield the exact value of the expected firsthitting time. Further, we also state the stationary distribution as well as the mixing time of a special case of our scenario.

Kötzing, Timo; Krejca, Martin S. FirstHitting Times Under Additive Drift. Parallel Problem Solving From Nature (PPSN) 2018: 92104
For the last ten years, almost every theoretical result concerning the expected run time of a randomized search heuristic used drift theory, making it the arguably most important tool in this domain. Its success is due to its ease of use and its powerful result: drift theory allows the user to derive bounds on the expected firsthitting time of a random process by bounding expected local changes of the process – the drift. This is usually far easier than bounding the expected firsthitting time directly. Due to the widespread use of drift theory, it is of utmost importance to have the best drift theorems possible. We improve the fundamental additive, multiplicative, and variable drift theorems by stating them in a form as general as possible and providing examples of why the restrictions we keep are still necessary. Our additive drift theorem for upper bounds only requires the process to be nonnegative, that is, we remove unnecessary restrictions like a finite, discrete, or bounded search space. As corollaries, the same is true for our upper bounds in the case of variable and multiplicative drift

Frahnow, Clemens; Kötzing, Timo Ring Migration Topology Helps Bypassing Local Optima. Parallel Problem Solving From Nature (PPSN) 2018: 129140
Running several evolutionary algorithms in parallel and occasionally exchanging good solutions is referred to as island models. The idea is that the independence of the different islands leads to diversity, thus possibly exploring the search space better. Many theoretical analyses so far have found a complete (or sufficiently quickly expanding) topology as underlying migration graph most efficient for optimization, even though a quick dissemination of individuals leads to a loss of diversity. We suggest a simple fitness function Fork with two local optima parametrized by \(r \geq 2\) and a scheme for composite fitness functions. We show that, while the (1+1) EA gets stuck in a bad local optimum and incurs a run time of \(\Theta(n^{2r})\) fitness evaluations on Fork, island models with a complete topology can achieve a run time of \(\Theta(n^{1.5r})\) by making use of rare migrations in order to explore the search space more effectively. Finally, the ring topology, making use of rare migrations and a large diameter, can achieve a run time of \(\tilde{\Theta}(n^r)\), the black box complexity of Fork. This shows that the ring topology can be preferable over the complete topology in order to maintain diversity.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. The Compact Genetic Algorithm is Efficient under Extreme Gaussian Noise. IEEE Transactions on Evolutionary Computation 2017: 477490
Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noisehandling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noisehandling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the compact Genetic Algorithm can overcome the shortsightedness of mutationonly heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.

Friedrich, Tobias; Kötzing, Timo; Wagner, Markus A Generic BetandRun Strategy for Speeding Up Stochastic Local Search. Conference on Artificial Intelligence (AAAI) 2017: 801807
A common strategy for improving optimization algorithms is to restart the algorithm when it is believed to be trapped in an inferior part of the search space. However, while specific restart strategies have been developed for specific problems (and specific algorithms), restarts are typically not regarded as a general tool to speed up an optimization algorithm. In fact, many optimization algorithms do not employ restarts at all. Recently, "betandrun" was introduced in the context of mixedinteger programming, where first a number of short runs with randomized initial conditions is made, and then the most promising run of these is continued. In this article, we consider two classical NPcomplete combinatorial optimization problems, traveling salesperson and minimum vertex cover, and study the effectiveness of different betandrun strategies. In particular, our restart strategies do not take any problem knowledge into account, nor are tailored to the optimization algorithm. Therefore, they can be used offtheshelf. We observe that stateoftheart solvers for these problems can benefit significantly from restarts on standard benchmark instances.

Kötzing, Timo; Schirneck, Martin; Seidel, Karen Normal Forms in Semantic Language Identification. Algorithmic Learning Theory (ALT) 2017: 493516
We consider language learning in the limit from text where all learning restrictions are semantic, that is, where any conjecture may be replaced by a semantically equivalent conjecture. For different such learning criteria, starting with the wellknown TxtGBclearning, we consider three different normal forms: strongly locking learning, consistent learning and (partially) setdriven learning. These normal forms support and simplify proofs and give insight into what behaviors are necessary for successful learning (for example when consistency in conservative learning implies cautiousness and strong decisiveness). We show that strongly locking learning can be assumed for partially setdriven learners, even when learning restrictions apply. We give a very general proof relying only on a natural property of the learning restriction, namely, allowing for simulation on equivalent text. Furthermore, when no restrictions apply, also the converse is true: every strongly locking learner can be made partially setdriven. For several semantic learning criteria we show that learning can be done consistently. Finally, we deduce for which learning restrictions partial setdrivenness and setdrivenness coincide, including a general statement about classes of infinite languages. The latter again relies on a simulation argument.

Gao, Wanru; Friedrich, Tobias; Kötzing, Timo; Neumann, Frank Scaling up Local Search for Minimum Vertex Cover in Large Graphs by Parallel Kernelization. Australasian Conference on Artificial Intelligence (AUSAI) 2017: 131143
We investigate how wellperforming local search algorithms for small or medium size instances can be scaled up to perform well for large inputs. We introduce a parallel kernelization technique that is motivated by the assumption that graphs in medium to large scale are composed of components which are on their own easy for stateoftheart solvers but when hidden in large graphs are hard to solve. To show the effectiveness of our kernelization technique, we consider the wellknown minimum vertex cover problem and two stateoftheart solvers called NuMVC and FastVC. Our kernelization approach reduces an existing large problem instance significantly and produces better quality results on a wide range of benchmark instances and real world graphs.

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.

Friedrich, Tobias; Kötzing, Timo; Melnichenko, Anna Analyzing Search Heuristics with Differential Equations. Genetic and Evolutionary Computation Conference (GECCO) 2017: 313314
Drift Theory is currently the most common technique for the analysis of randomized search heuristics because of its broad applicability and the resulting tight first hitting time bounds. The biggest problem when applying a drift theorem is to find a suitable potential function which maps a complex space into a single number, capturing the essence of the state of the search in just one value. We discuss another method for the analysis of randomized search heuristics based on the Theory of Differential Equations. This method considers the deterministic counterpart of the randomized process by replacing probabilistic outcomes by their expectation, and then bounding the error with good probability. We illustrate this by analyzing an Ant Colony Optimization algorithm (ACO) for the Minimum Spanning Tree problem (MST).

Doerr, Benjamin; Kötzing, Timo; Lagodzinski, J. A. Gregor; Lengler, Johannes Bounding Bloat in Genetic Programming. Genetic and Evolutionary Computation Conference (GECCO) 2017: 921928
While many optimization problems work with a fixed number of decision variables and thus a fixedlength representation of possible solutions, genetic programming (GP) works on variablelength representations. A naturally occurring problem is that of bloat (unnecessary growth of solutions) slowing down optimization. Theoretical analyses could so far not bound bloat and required explicit assumptions on the magnitude of bloat. In this paper we analyze bloat in mutationbased genetic programming for the two test functions ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat and give matching or closetomatching upper and lower bounds for the expected optimization time. In particular, we show that the \((1+1)\) GP takes (i) \(\Theta(T_\text{init + n\, \log n)\) iterations with bloat control on ORDER as well as MAJORITY; and (ii) \(O(T_{\text{init ,log \,T_{\text{init + n(\log n)^3)\) and \(\Omega(T_\text{init + n \,\log n)\) (and \(\Omega(T_\text{init \,\log \,T_{\text{init}})\) for \(n = 1\)) iterations without bloat control on MAJORITY.

Doerr, Benjamin; Fischbeck, Philipp; Frahnow, Clemens; Friedrich, Tobias; Kötzing, Timo; Schirneck, Martin Island Models Meet Rumor Spreading. Genetic and Evolutionary Computation Conference (GECCO) 2017: 13591366
Island models in evolutionary computation solve problems by a careful interplay of independently running evolutionary algorithms on the island and an exchange of good solutions between the islands. In this work, we conduct rigorous run time analyses for such island models trying to simultaneously obtain good run times and low communication effort. We improve the existing upper bounds for the communication effort (i) by improving the run time bounds via a careful analysis, (ii) by setting the balance between individual computation and communication in a more appropriate manner, and (iii) by replacing the usual communicatewithallneighbors approach with randomized rumor spreading, where each island contacts a randomly chosen neighbor. This epidemic communication paradigm is known to lead to very fast and robust information dissemination in many applications. Our results concern islands running simple (1+1) evolutionary algorithms, we regard ddimensional tori and complete graphs as communication topologies, and optimize the classic test functions OneMax and LeadingOnes.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Unknown Solution Length Problems With No Asymptotically Optimal Run Time. Genetic and Evolutionary Computation Conference (GECCO) 2017: 13671374
We revisit the problem of optimizing a fitness function of unknown dimension; that is, we face a function defined over bitstrings of large length \(N\), but only \(n << N\) of them have an influence on the fitness. Neither the position of these relevant bits nor their number is known. In previous work, variants of the \((1 + 1)\) evolutionary algorithm (EA) have been developed that solve, for arbitrary \(s \in N\), such OneMax and LeadingOnes instances, simultaneously for all \(n \in N\), in expected time \(O(n(\log(n))^2 log \log(n) dots \log^{s1 (n)(\log^{(s) (n))^1+\epsilon} )\)and \(O(n^2 \log(n) log \log(n) dots \log^{s1 (n)(\log^{(s) (n))^1+\epsilon} )\), respectively; that is, in almost the same time as if n and the relevant bit positions were known. In this work, we prove the first, almost matching, lower bounds for this setting. For LeadingOnes, we show that, for every \(s \in N\), the \((1 + 1)\) EA with any mutation operator treating zeros and ones equally has an expected run time of \(\omega(n^2 \log(n) log \log(n) dots \log^{(s) (n))\) when facing problem size \(n\). Aiming at closing the small remaining gap, we realize that, quite surprisingly, there is no asymptotically best performance. For any algorithm solving, for all \(n\), all instances of size \(n\) in expected time at most \(T (n)\), there is an algorithm doing the same in time \(T'(n)\) with \(T' = o(T )\). For OneMax we show results of similar flavor.

Shi, Feng; Schirneck, Martin; Friedrich, Tobias; Kötzing, Timo; Neumann, Frank Reoptimization Times of Evolutionary Algorithms on Linear Functions Under Dynamic Uniform Constraints. Genetic and Evolutionary Computation Conference (GECCO) 2017: 14071414
Thee investigations of linear pseudoBoolean functions play a central role in the area of runtime analysis of evolutionary computing techniques. Having an additional linear constraint on a linear function is equivalent to the NPhard knapsack problem and special problem classes thereof have been investigated in recent works. In this paper, we extend these studies to problems with dynamic constraints and investigate the runtime of different evolutionary algorithms to recompute an optimal solution when the constraint bound changes by a certain amount. We study the classical \((1+1)\) EA and populationbased algorithms and show that they recompute an optimal solution very efficiently. Furthermore, we show that a variant of the \((1+(\lambda, \lambda))\) GA can recompute the optimal solution more efficiently in some cases.

Gießen, Christian; Kötzing, Timo Robustness of Populations in Stochastic Environments. Algorithmica 2016: 462489
We consider stochastic versions of OneMax and LeadingOnes and analyze the performance of evolutionary algorithms with and without populations on these problems. It is known that the (1+1) EA on OneMax performs well in the presence of very small noise, but poorly for higher noise levels. We extend these results to LeadingOnes and to many different noise models, showing how the application of drift theory can significantly simplify and generalize previous analyses. Most surprisingly, even small populations (of size \(\Theta(\log n))\) can make evolutionary algorithms perform well for high noise levels, well outside the abilities of the (1+1) EA. Larger population sizes are even more beneficial; we consider both parent and offspring populations. In this sense, populations are robust in these stochastic settings.

Kötzing, Timo Concentration of First Hitting Times Under Additive Drift. Algorithmica 2016: 490506
Recent advances in drift analysis have given us better and better tools for understanding random processes, including the run time of randomized search heuristics. In the setting of multiplicative drift we do not only have excellent bounds on the expected run time, but also more general results showing the concentration of the run time. In this paper we investigate the setting of additive drift under the assumption of strong concentration of the "step size" of the process. Under sufficiently strong drift towards the goal we show a strong concentration of the hitting time. In contrast to this, we show that in the presence of small drift a Gambler'sRuinlike behavior of the process overrides the influence of the drift. Finally, in the presence of sufficiently strong negative drift the hitting time is superpolynomial with high probability; this corresponds to the socalled negative drift theorem, for which we give new variants.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. Robustness of Ant Colony Optimization to Noise. Evolutionary Computation 2016: 237254
Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding. We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with additive posterior noise sampled from a Gaussian distribution. Without noise the classical \((\mu+1)\)EA outperforms any ACO algorithm, with smaller \(\mu\) being better; however, with large noise, the \((\mu+1)\)EA fails, even for high values of \(\mu\) (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor \(p\) is small enough dependent on the parameter \(\delta^2\) of the noise and the dimension \(n\) of the search space \((p = o(1/(n(n + \delta \log n)^2 \log n)))\), optimization will be successful.

Case, John; Kötzing, Timo Strongly nonUshaped language learning results by general techniques. Information and Computation 2016: 115
In learning, a semantic or behavioral Ushape occurs when a learner first learns, then unlearns, and, finally, relearns, some target concept. This paper introduces two general techniques and applies them especially to syntactic Ushapes in learning: one technique to show when they are necessary and one to show when they are unnecessary. The technique for the former is very general and applicable to a much wider range of learning criteria. It employs socalled selflearning classes of languages which are shown to characterize completely one criterion learning more than another. We apply these techniques to show that, for setdriven and rearrangementindependent learning, any kind of Ushapes is unnecessary. Furthermore, we show that Ushapes are necessary in a strong way for iterative learning, contrasting with an earlier result by Case and Moelius that semantic Ushapes are unnecessary for iterative learning.

Jain, Sanjay; Kötzing, Timo; Ma, Junqi; Stephan, Frank On the Role of Update Constraints and TextTypes in Iterative Learning. Information and Computation 2016: 152168
The present work investigates the relationship of iterative learning with other learning criteria such as decisiveness, caution, reliability, nonUshapedness, monotonicity, strong monotonicity and conservativeness. Building on the result of Case and Moelius that iterative learners can be made nonUshaped, we show that they also can be made cautious and decisive. Furthermore, we obtain various special results with respect to oneone texts, fat texts and oneone hypothesis spaces.

Jain, Sanjay; Kötzing, Timo; Stephan, Frank Enlarging learnable classes. Information and Computation 2016: 194207
We study which classes of recursive functions satisfy that their union with any other explanatorily learnable class of recursive functions is again explanatorily learnable. We provide sufficient criteria for classes of recursive functions to satisfy this property and also investigate its effective variants. Furthermore, we study the question which learners can be effectively extended to learn a larger class of functions. We solve an open problem by showing that there is no effective procedure which does this task on all learners which do not learn a dense class of recursive functions. However, we show that there are two effective extension procedures such that each learner is extended by one of them.

Kötzing, Timo; Palenta, Raphaela A map of update constraints in inductive inference. Theoretical Computer Science 2016: 424
We investigate how different learning restrictions reduce learning power and how the different restrictions relate to one another. We give a complete map for nine different restrictions both for the cases of complete information learning and setdriven learning. This completes the picture for these wellstudied delayable learning restrictions. A further insight is gained by different characterizations of conservative learning in terms of variants of cautious learning. Our analyses greatly benefit from general theorems we give, for example showing that learners with exclusively delayable restrictions can always be assumed total.

Case, John; Kötzing, Timo Topological Separations in Inductive Inference. Theoretical Computer Science 2016: 3345
Re learning in the limit from positive data, a major concern is which classes of languages are learnable with respect to a given learning criterion. We are particularly interested herein in the reasons for a class of languages to be unlearnable. We consider two types of reasons. One type is called topological where it does not help if the learners are allowed to be uncomputable (an example of Gold's is that no class containing an infinite language and all its finite sublanguages is learnable  even by an uncomputable learner). Another reason is called computational (where the learners are required to be algorithmic). In particular, two learning criteria might allow for learning different classes of languages from one another  but with dependence on whether the unlearnability is of type topological or computational. In this paper we formalize the idea of two learning criteria separating topologically in learning power. This allows us to study more closely why two learning criteria separate in learning power. For a variety of learning criteria, concerning vacillatory, monotone, (several kinds of) iterative and feedback learning, we show that certain learning criteria separate topologically, and certain others, which are known to separate, are shown not to separate topologically. Showing that learning criteria do not separate topologically implies that any known separation must necessarily exploit algorithmicity of the learner.

Friedrich, Tobias; Kötzing, Timo; Quinzan, Francesco; Sutton, Andrew M. Ant Colony Optimization Beats Resampling on Noisy Functions. Genetic and Evolutionary Computation Conference (GECCO) 2016: 34
Despite the pervasiveness of noise in realworld optimization, there is little understanding of the interplay between the operators of randomized search heuristics and explicit noisehandling techniques such as statistical resampling. Ant Colony Optimization (ACO) algorithms are claimed to be particularly wellsuited to dynamic and noisy problems, even without explicit noisehandling techniques. In this work, we empirically investigate the tradeoffs between resampling an the noisehandling abilities of ACO algorithms. Our main focus is to locate the point where resampling costs more than it is worth.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. The Benefit of Recombination in Noisy Evolutionary Search. Genetic and Evolutionary Computation Conference (GECCO) 2016: 161162
Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noisehandling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noisehandling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the Compact Genetic Algorithm can overcome the shortsightedness of mutationonly heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.

Dang, DucCuong; Friedrich, Tobias; Krejca, Martin S.; Kötzing, Timo; Lehre, Per Kristian; Oliveto, Pietro S.; Sudholt, Dirk; Sutton, Andrew Michael Escaping Local Optima with Diversity Mechanisms and Crossover. Genetic and Evolutionary Computation Conference (GECCO) 2016: 645652
Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the \((\mu+1)\) GA using uniform crossover on the fitness function \(Jump_k\). All previous results in this context only hold for unrealistically low crossover probability \(p_c=O(k/n)\), while we give analyses for the setting of constant \(p_c < 1\) in all but one case. Our bounds show a dependence on the problem size \(n\), the jump length \(k\), the population size \(\mu\), and the crossover probability \(p_c\). For the typical case of constant \(k > 2\) and constant \(p_c\), we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of \(\mu\): \(O(n^{k1})\) for duplicate elimination/minimisation, \(O(n^2 \log n)\) for maximising the convex hull, \(O(n \log n)\) for det. crowding (assuming \(p_c = k/n\)), \(O(n \log n)\) for maximising the Hamming distance, \(O(n \log n)\) for fitness sharing, \(O(n \log n)\) for the singlereceiver island model. This proves a sizeable advantage of all variants of the \((\mu+1)\) GA compared to the (1+1) EA, which requires \(\Theta(n^k)\). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Nallaperuma, Samadhi; Neumann, Frank; Schirneck, Martin Fast Building Block Assembly by Majority Vote Crossover. Genetic and Evolutionary Computation Conference (GECCO) 2016: 661668
Different works have shown how crossover can help with building block assembly. Typically, crossover might get lucky to select good building blocks from each parent, but these lucky choices are usually rare. In this work we consider a crossover operator which works on three parent individuals. In each component, the offspring inherits the value present in the majority of the parents; thus, we call this crossover operator majority vote. We show that, if good components are sufficiently prevalent in the individuals, majority vote creates an optimal individual with high probability. Furthermore, we show that this process can be amplified: as long as components are good independently and with probability at least \(1/2+\delta\), we require only \(O(\log 1/\delta + \log \log n)\) successive stages of majority vote to create an optimal individual with high probability! We show how this applies in two scenarios. The first scenario is the Jump test function. With sufficient diversity, we get an optimization time of \(O(n \log n)\) even for jump sizes as large as \(O(n^{(1/2\epsilon)})\). Our second scenario is a family of vertex cover instances. Majority vote optimizes this family efficiently, while local searches fail and only highly specialized twoparent crossovers are successful.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo The Right Mutation Strength for MultiValued Decision Variables. Genetic and Evolutionary Computation Conference (GECCO) 2016: 11151122
The most common representation in evolutionary computation are bit strings. This is ideal to model binary decision variables, but less useful for variables taking more values. With very little theoretical work existing on how to use evolutionary algorithms for such optimization problems, we study the run time of simple evolutionary algorithms on some OneMaxlike functions defined over \(\Omega=\{0,1,\dots,r1\}n\). More precisely, we regard a variety of problem classes requesting the componentwise minimization of the distance to an unknown target vector \(z \in \Omega\). For such problems we see a crucial difference in how we extend the standardbit mutation operator to these multivalued domains. While it is natural to select each position of the solution vector to be changed independently with probability \(1/n\), there are various ways to then change such a position. If we change each selected position to a random value different from the original one, we obtain an expected run time of \(\Theta(nr\log n)\). If we change each selected position by either +1 or 1 (random choice), the optimization time reduces to \(\Theta(nr+n\log n)\). If we use a random mutation strength \(i \in \{0,1,\dots,r1\}n\) with probability inversely proportional to \(i\) and change the selected position by either +\(i\) or \(i\) (random choice), then the optimization time becomes \(\Theta(n\log(r)(\log(n)+\log(r)))\), bringing down the dependence on \(r\) from linear to polylogarithmic. One of our results depends on a new variant of the lower bounding multiplicative drift theorem.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S. EDAs cannot be Balanced and Stable. Genetic and Evolutionary Computation Conference (GECCO) 2016: 11391146
Estimation of Distribution Algorithms (EDAs) work by iteratively updating a distribution over the search space with the help of samples from each iteration. Up to now, theoretical analyses of EDAs are scarce and present run time results for specific EDAs. We propose a new framework for EDAs that captures the idea of several known optimizers, including PBIL, UMDA, \(\lambda\)MMASIB, cGA, and \((1,\lambda)\)EA. Our focus is on analyzing two core features of EDAs: a balanced EDA is sensitive to signals in the fitness; a stable EDA remains uncommitted under a biasless fitness function. We prove that no EDA can be both balanced and stable. The LeadingOnes function is a prime example where, at the beginning of the optimization, the fitness function shows no bias for many bits. Since many wellknown EDAs are balanced and thus not stable, they are not wellsuited to optimize LeadingOnes. We give a stable EDA which optimizes LeadingOnes within a time of \(O(n\,\log n)\).

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. Graceful Scaling on Uniform versus SteepTailed Noise. Parallel Problem Solving From Nature (PPSN) 2016: 761770
Recently, different evolutionary algorithms (EAs) have been analyzed in noisy environments. The most frequently used noise model for this was additive posterior noise (noise added after the fitness evaluation) taken from a Gaussian distribution. In particular, for this setting it was shown that the \((\mu + 1)\)EA on OneMax does not scale gracefully (higher noise cannot efficiently be compensated by higher \(\mu\)). In this paper we want to understand whether there is anything special about the Gaussian distribution which makes the \((\mu + 1)\)EA not scale gracefully. We keep the setting of posterior noise, but we look at other distributions. We see that for exponential tails the \((\mu + 1)\)EA on OneMax does also not scale gracefully, for similar reasons as in the case of Gaussian noise. On the other hand, for uniform distributions (as well as other, similar distributions) we see that the \((\mu + 1)\)EA on OneMax does scale gracefully, indicating the importance of the noise model.

Friedrich, Tobias; Kötzing, Timo; Sutton, Andrew M. On the Robustness of Evolving Populations. Parallel Problem Solving From Nature (PPSN) 2016: 771781
Most theoretical work that studies the benefit of recombination focuses on the ability of crossover to speed up optimization time on specific search problems. In this paper, we take a slightly different perspective and investigate recombination in the context of evolving solutions that exhibit \(\emph{mutational}\) robustness, i.e., they display insensitivity to small perturbations. Various models in population genetics have demonstrated that increasing the effective recombination rate promotes the evolution of robustness. We show this result also holds in the context of evolutionary computation by proving crossover promotes the evolution of robust solutions in the standard \((\mu+1)\) GA. Surprisingly, our results show that the effect is present even when robust solutions are at a selective disadvantage due to lower fitness values.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Provably Optimal SelfAdjusting Step Sizes for MultiValued Decision Variables. Parallel Problem Solving From Nature (PPSN) 2016: 782791
We regard the problem of maximizing a OneMaxlike function defined over an alphabet of size \(r\). In previous work [GECCO 2016] we have investigated how three different mutation operators influence the performance of Randomized Local Search (RLS) and the (1+1) Evolutionary Algorithm. This work revealed that among these natural mutation operators none is superior to the other two for any choice of \(r\). We have also given in [GECCO 2016] some indication that the best achievable run time for large \(r\) is \(\Theta(n log r(\log n + \log r))\), regardless of how the mutation operator is chosen, as long as it is a static choice (i.e., the distribution used for variation of the current individual does not change over time). Here in this work we show that we can achieve a better performance if we allow for adaptive mutation operators. More precisely, we analyze the performance of RLS using a selfadjusting mutation strength. In this algorithm the size of the steps taken in each iteration depends on the success of previous iterations. That is, the mutation strength is increased after a successful iteration and it is decreased otherwise. We show that this idea yields an expected optimization time of \(\Theta(n(\log n + \log r))\), which is optimal among all comparisonbased search heuristics. This is the first time that selfadjusting parameter choices are shown to outperform static choices on a discrete multivalued optimization problem.

Dang, DucCuong; Lehre, Per Kristian; Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Oliveto, Pietro S.; Sudholt, Dirk; Sutton, Andrew M. Emergence of Diversity and its Benefits for Crossover in Genetic Algorithms. Parallel Problem Solving From Nature (PPSN) 2016: 890900
Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for a standard \((\mu+1)\) GA and the \(Jump_k\) test function. By studying the stochastic process underlying the size of the largest collection of identical genotypes we show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order \(\Omega(n/ \log n)\) compared to mutationonly algorithms like the \((1+1)\) EA.

Kötzing, Timo; Schirneck, Martin Towards an Atlas of Computational Learning Theory. Symposium on Theoretical Aspects of Computer Science (STACS) 2016: 47:147:13
A major part of our knowledge about Computational Learning stems from comparisons of the learning power of different learning criteria. These comparisons inform about tradeoffs between learning restrictions and, more generally, learning settings; furthermore, they inform about what restrictions can be observed without losing learning power. With this paper we propose that one main focus of future research in Computational Learning should be on a structured approach to determine the relations of different learning criteria. In particular, we propose that, for small sets of learning criteria, all pairwise relations should be determined; these relations can then be easily depicted as a map, a diagram detailing the relations. Once we have maps for many relevant sets of learning criteria, the collection of these maps is an Atlas of Computational Learning Theory, informing at a glance about the landscape of computational learning just as a geographical atlas informs about the earth. In this paper we work toward this goal by providing three example maps, one pertaining to partially setdriven learning, and two pertaining to strongly monotone learning. These maps can serve as blueprints for future maps of similar base structure.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Unbiased BlackBox Complexities of Jump Functions. Evolutionary Computation 2015: 641670
We analyze the unbiased blackbox complexity of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is \((1/2\epsilon)n\), that is, only a small constant fraction of the fitness values is visible, then the unbiased blackbox complexities for arities 3 and higher are of the same order as those for the simple OneMax function. Even for the extreme jump function, in which all but the two fitness values \(n/2\) and \(n\) are blanked out, polynomialtime mutationbased (i.e., unary unbiased) blackbox optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a \(\Theta(n^{1/2})\) fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased blackbox complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.

Freydenberger, Dominik D.; Kötzing, Timo Fast Learning of Restricted Regular Expressions and DTDs. Theory of Computing Systems 2015: 11141158
We study the problem of generalizing from a finite sample to a language taken from a predefined language class. The two language classes we consider are subsets of the regular languages and have significance in the specification of XML documents (the classes corresponding to so called chain regular expressions, Chares, and to single occurrence regular expressions, Sores). The previous literature gave a number of algorithms for generalizing to Sores providing a trade off between quality of the solution and speed. Furthermore, a fast but nonoptimal algorithm for generalizing to Chares is known. For each of the two language classes we give an efficient algorithm returning a minimal generalization from the given finite sample to an element of the fixed language class; such generalizations are called descriptive. In this sense, both our algorithms are optimal.

Kötzing, Timo; Lissovoi, Andrei; Witt, Carsten (1+1) EA on Generalized Dynamic OneMax. Foundations of Genetic Algorithms (FOGA) 2015: 4051
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 reprove 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.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. Robustness of Ant Colony Optimization to Noise. Genetic and Evolutionary Computation Conference (GECCO) 2015: 1724
Best Paper Award (ACO/SI Track)
Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding. We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with additive posterior noise sampled from a Gaussian distribution. Without noise the classical \((\mu+1)\)EA outperforms any ACO algorithm, with smaller \(\mu\) being better; however, with large noise, the \((\mu+1)\)EA fails, even for high values of \(\mu\) (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor \(\mu\) is small enough dependent on the parameter \(\delta^2\) of the noise and the dimension \(n\) of the search space \((p = o(1/(n(n + \delta \log n)^2 \log n)))\), optimization will be successful.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Solving Problems with Unknown Solution Length at (Almost) No Extra Cost. Genetic and Evolutionary Computation Conference (GECCO) 2015: 831838
Most research in the theory of evolutionary computation assumes that the problem at hand has a fixed problem size. This assumption does not always apply to realworld optimization challenges, where the length of an optimal solution may be unknown a priori. Following up on previous work of Cathabard, Lehre, and Yao [FOGA 2011] we analyze variants of the (1+1) evolutionary algorithm for problems with unknown solution length. For their setting, in which the solution length is sampled from a geometric distribution, we provide mutation rates that yield an expected optimization time that is of the same order as that of the (1+1) EA knowing the solution length. We then show that almost the same run times can be achieved even if no a priori information on the solution length is available. Finally, we provide mutation rates suitable for settings in which neither the solution length nor the positions of the relevant bits are known. Again we obtain almost optimal run times for the OneMax and LeadingOnes test functions, thus solving an open problem from Cathabard et al.

Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. The Benefit of Recombination in Noisy Evolutionary Search. International Symposium of Algorithms and Computation (ISAAC) 2015: 140150
Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings metaheuristics are a popular choice for deriving good optimization algorithms, most notably evolutionary algorithms which mimic evolution in nature. Empirical evidence suggests that genetic recombination is useful in uncertain environments because it can stabilize a noisy fitness signal. With this paper we want to support this claim with mathematical rigor. The setting we consider is that of noisy optimization. We study a simple noisy fitness function that is derived by adding Gaussian noise to a monotone function. First, we show that a classical evolutionary algorithm that does not employ sexual recombination (the \((\mu+1)\)EA) cannot handle the noise efficiently, regardless of the population size. Then we show that an evolutionary algorithm which does employ sexual recombination (the Compact Genetic Algorithm, short: cGA) can handle the noise using a graceful scaling of the population.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo The unbiased blackbox complexity of partition is polynomial. Artificial Intelligence 2014: 275286
Unbiased blackbox complexity was introduced as a refined complexity model for randomized search heuristics (Lehre and Witt (2012) [24]). For several problems, this notion avoids the unrealistically low complexity results given by the classical model of Droste et al. (2006) [10]. We show that for some problems the unbiased blackbox complexity remains artificially small. More precisely, for two different formulations of an NP hard subclass of the wellknown Partition problem, we give mutationonly unbiased blackbox algorithms having complexity \(O(n \log n)\). This indicates that also the unary unbiased blackbox complexity does not give a complete picture of the true difficulty of this problem for randomized search heuristics.

Kötzing, Timo; Sutton, Andrew M.; Neumann, Frank; O'Reilly, UnaMay The Max problem revisited: The importance of mutation in genetic programming. Theoretical Computer Science 2014: 94107
This paper contributes to the rigorous understanding of genetic programming algorithms by providing runtime complexity analyses of the wellstudied Max problem. Several experimental studies have indicated that it is hard to solve the Max problem with crossoverbased algorithms. Our analyses show that different variants of the Max problem can provably be solved using simple mutationbased genetic programming algorithms. Our results advance the body of computational complexity analyses of genetic programming, indicate the importance of mutation in genetic programming, and reveal new insights into the behavior of mutationbased genetic programming algorithms.

Kötzing, Timo Iterative learning from positive data and counters. Theoretical Computer Science 2014: 155169
We analyze iterative learning in the limit from positive data with the additional information provided by a counter. The simplest type of counter provides the current iteration number (counting up from 0 to infinity), which is known to improve learning power over plain iterative learning. We introduce five other (weaker) counter types, for example only providing some unbounded and nondecreasing sequence of numbers. Analyzing these types allows for understanding what properties of a counter can benefit learning. For the iterative setting, we completely characterize the relative power of the learning criteria corresponding to the counter types. In particular, for our types, the only properties improving learning power are unboundedness and strict monotonicity. Furthermore, we show that each of our types of counter improves learning power over weaker ones in some settings, and that, for iterative learning criteria with one of these types of counter, separations of learning criteria are necessarily witnessed by classes containing only infinite languages.

Kötzing, Timo; Palenta, Raphaela A Map of Update Constraints in Inductive Inference. Algorithmic Learning Theory (ALT) 2014: 4054
We investigate how different learning restrictions reduce learning power and how the different restrictions relate to one another. We give a complete map for nine different restrictions both for the cases of complete information learning and setdriven learning. This completes the picture for these wellstudied delayable learning restrictions. A further insight is gained by different characterizations of conservative learning in terms of variants of cautious learning. Our analyses greatly benefit from general theorems we give, for example showing that learners with exclusively delayable restrictions can always be assumed total.

Jain, Sanjay; Kötzing, Timo; Ma, Junqi; Stephan, Frank On the Role of Update Constraints and TextTypes in Iterative Learning. Algorithmic Learning Theory (ALT) 2014: 5569
The present work investigates the relationship of iterative learning with other learning criteria such as decisiveness, caution, reliability, nonUshapedness, monotonicity, strong monotonicity and conservativeness. Building on the result of Case and Moelius that iterative learners can be made nonUshaped, we show that they also can be made cautious and decisive. Furthermore, we obtain various special results with respect to oneone texts, fat texts and oneone hypothesis spaces.

Doerr, Benjamin; Doerr, Carola; Kötzing, Timo Unbiased blackbox complexities of jump functions: how to cross large plateaus. Genetic and Evolutionary Computation Conference (GECCO) 2014: 769776
We analyze the unbiased blackbox complexity of jump functions with large jump sizes. Among other results, we show that when the jump size is \((1/2  \epsilon)n\), that is, only a small constant fraction of the fitness values is visible, then the unbiased blackbox complexities for arities 3 and higher are of the same order as those for the simple OneMax function. Even for the extreme jump function, in which all but the two fitness values \(n/2\) and \(n\) are blanked out, polynomialtime mutationbased (i.e., unary unbiased) blackbox optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a \(\Theta(n^{1/2})\) fraction) is a plateau of constant fitness.To prove these results, we introduce new tools for the analysis of unbiased blackbox complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.

Gießen, Christian; Kötzing, Timo Robustness of populations in stochastic environments. Genetic and Evolutionary Computation Conference (GECCO) 2014: 13831390
We consider stochastic versions of OneMax and LeadingOnes and analyze the performance of evolutionary algorithms with and without populations on these problems. It is known that the (1+1) EA on OneMax performs well in the presence of very small noise, but poorly for higher noise levels. We extend these results to LeadingOnes and to many different noise models, showing how the application of drift theory can significantly simplify and generalize previous analyses. Most surprisingly, even small populations (of size \(\Theta(\log n)\)) can make evolutionary algorithms perform well for high noise levels, well outside the abilities of the (1+1) EA. Larger population sizes are even more beneficial; we consider both parent and offspring populations. In this sense, populations are robust in these stochastic settings.

Kötzing, Timo Concentration of first hitting times under additive drift. Genetic and Evolutionary Computation Conference (GECCO) 2014: 13911398
Recent advances in drift analysis have given us better and better tools for understanding random processes, including the run time of randomized search heuristics. In the setting of multiplicative drift we do not only have excellent bounds on the expected run time, but also more general results showing the concentration of the run time. In this paper we investigate the setting of additive drift under the assumption of strong concentration of the "step size" of the process. Under sufficiently strong drift towards the goal we show a strong concentration of the hitting time. In contrast to this, we show that in the presence of small drift a Gambler'sRuinlike behavior of the process overrides the influence of the drift. Finally, in the presence of sufficiently strong negative drift the hitting time is superpolynomial with high probability; this corresponds to the socalled negative drift theorem, for which we give new variants.

Kötzing, Timo A Solution to Wiehagen's Thesis. Symposium on Theoretical Aspects of Computer Science (STACS) 2014: 494505
Wiehagen's Thesis in Inductive Inference (1991) essentially states that, for each learning criterion, learning can be done in a normalized, enumerative way. The thesis was not a formal statement and thus did not allow for a formal proof, but support was given by examples of a number of different learning criteria that can be learned by enumeration. Building on recent formalizations of learning criteria, we are now able to formalize Wiehagen's Thesis. We prove the thesis for a wide range of learning criteria, including many popular criteria from the literature. We also show the limitations of the thesis by giving four learning criteria for which the thesis does not hold (and, in two cases, was probably not meant to hold). Beyond the original formulation of the thesis, we also prove stronger versions which allow for many corollaries relating to strongly decisive and conservative learning.

Doerr, Benjamin; Johannsen, Daniel; Kötzing, Timo; Neumann, Frank; Theile, Madeleine More effective crossover operators for the allpairs shortest path problem. Theoretical Computer Science 2013: 1226
The allpairs shortest path problem is the first nonartificial problem for which it was shown that adding crossover can significantly speed up a mutationonly 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 \(\Theta(n^3.25(\log n)^{0.25})\). 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 \(\emph{repair mechanisms}\) and \(\emph{parent selection}\). We show that repairing infeasible offspring leads to an improved expected optimization time of \(O(n^{3.2(\log n)^{0.2})\). As a second part of our study we prove that choosing parents that guarantee feasible offspring results in an even better optimization time of \(O(n^3 \log n)\). Both results show that already simple adjustments of the recombination operator can asymptotically improve the runtime of evolutionary algorithms.

Doerr, Benjamin; Kötzing, Timo; Lengler, Johannes; Winzen, Carola Blackbox complexities of combinatorial problems. Theoretical Computer Science 2013: 84106
Blackbox complexity is a complexity theoretic measure for how difficult a problem is to be optimized by a general purpose optimization algorithm. It is thus one of the few means trying to understand which problems are tractable for genetic algorithms and other randomized search heuristics. Most previous work on blackbox complexity is on artificial test functions. In this paper, we move a step forward and give a detailed analysis for the two combinatorial problems minimum spanning tree and singlesource shortest paths. Besides giving interesting bounds for their blackbox complexities, our work reveals that the choice of how to model the optimization problem is nontrivial here. This in particular comes true where the search space does not consist of bit strings and where a reasonable definition of unbiasedness has to be agreed on.

Case, John; Kötzing, Timo Memorylimited nonUshaped learning with solved open problems. Theoretical Computer Science 2013: 100123
In empirical cognitive science, for human learning, a semantic or behavioral Ushape occurs when a learner first learns, then unlearns, and, finally, relearns, some target concept. Within the formal framework of Inductive Inference, for learning from positive data, previous results have shown, for example, that such Ushapes are unnecessary for explanatory learning, but are necessary for behaviorally correct and nontrivial vacillatory learning. Herein we also distinguish between semantic and syntactic Ushapes. We answer a number of open questions in the prior literature as well as provide new results regarding syntactic Ushapes. Importantly for cognitive science, we see more of a previously noticed pattern that, for parameterized learning criteria, beyond very few initial parameter values, Ushapes are necessary for full learning power. We analyze the necessity of Ushapes in two memorylimited settings. The first setting is Bounded Memory State (BMS) learning, where a learner has an explicitlybounded state memory, and otherwise only knows its current datum. We show that there are classes learnable with three (or more) memory states that are not learnable nonUshapedly with any finite number of memory states. This result is surprising, since, for learning with one or two memory states, Ushapes are known to be unnecessary. This solves an open question from the literature. The second setting is that of Memoryless Feedback (MLF) learning, where a learner may ask a bounded number of questions about what data has been seen so far, and otherwise only knows its current datum. We show that there is a class learnable memorylessly with a single feedback query such that this class is not learnable nonUshapedly memorylessly with any finite number of feedback queries. We employ selflearning classes together with the Operator Recursion Theorem for many of our results, but we also introduce two new techniques for obtaining results. The first is for transferring inclusion results from one setting to another. The main part of the second is the Hybrid Operator Recursion Theorem, which enables us to separate some learning criteria featuring complexitybounded learners, employing selflearning classes. Both techniques are not specific to Ushaped learning, but applicable for a wide range of settings.

Case, John; Kötzing, Timo Topological Separations in Inductive Inference. Algorithmic Learning Theory (ALT) 2013: 128142
Re learning in the limit from positive data, a major concern is which classes of languages are learnable with respect to a given learning criterion. We are particularly interested herein in the reasons for a class of languages to be unlearnable. We consider two types of reasons. One type is called topological where it does not help if the learners are allowed to be uncomputable (an example of Gold's is that no class containing an infinite language and all its finite sublanguages is learnable  even by an uncomputable learner). Another reason is called computational (where the learners are required to be algorithmic). In particular, two learning criteria might allow for learning different classes of languages from one another  but with dependence on whether the unlearnability is of type topological or computational. In this paper we formalize the idea of two learning criteria separating topologically in learning power. This allows us to study more closely why two learning criteria separate in learning power. For a variety of learning criteria, concerning vacillatory, monotone, (several kinds of) iterative and feedback learning, we show that certain learning criteria separate topologically, and certain others, which are known to separate, are shown not to separate topologically. Showing that learning criteria do not separate topologically implies that any known separation must necessarily exploit algorithmicity of the learner.

Feldmann, Matthias; Kötzing, Timo Optimizing expected path lengths with ant colony optimization using fitness proportional update. Foundations of Genetic Algorithms (FOGA) 2013: 6574
We study the behavior of a MaxMin Ant System (MMAS) on the stochastic singledestination shortest path (SDSP) problem. Two previous papers already analyzed this setting for two slightly different MMAS algorithms, where the pheromone update fitnessindependently rewards edges of the bestsofar solution. The first paper showed that, when the bestsofar 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 noisefree weights. The second paper used reevaluation of the bestsofar 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 fitnessproportional update on stochasticweight graphs with arbitrary random edge weights from \([0,1]\). For \(\delta\) such that any suboptimal path is worse by at least \(\delta\) than an optimal path, then, with suitable parameters, the graph will be optimized after \(O(n^3 ln n/\delta over \delta^3)\) iterations (in expectation). In order to prove the above result, the multiplicative and the variable drift theorem are adapted to continuous search spaces.

Benz, Florian; Kötzing, Timo An effective heuristic for the smallest grammar problem. Genetic and Evolutionary Computation Conference (GECCO) 2013: 487494
The smallest grammar problem is the problem of finding the smallest contextfree grammar that generates exactly one given sequence. Approximating the problem with a ratio of less than 8569/8568 is known to be NPhard. Most work on this problem has focused on finding decent solutions fast (mostly in linear time), rather than on good heuristic algorithms. Inspired by a new perspective on the problem presented by Carrascosa et al. (2010), we investigate the performance of different heuristics on the problem. The aim is to find a good solution on large instances by allowing more than linear time. We propose a hybrid of a maxmin ant system and a genetic algorithm that in combination with a novel local search outperforms the state of the art on all files of the Canterbury corpus, a standard benchmark suite. Furthermore, this hybrid performs well on a standard DNA corpus.

Freydenberger, Dominik D.; Kötzing, Timo Fast learning of restricted regular expressions and DTDs. International Conference on Database Theory (ICDT) 2013: 4556
We study the problem of generalizing from a finite sample to a language taken from a predefined language class. The two language classes we consider are subsets of the regular languages and have significance in the specification of XML documents (the classes corresponding to so called chain regular expressions, Chares, and to single occurrence regular expressions, Sores). The previous literature gave a number of algorithms for generalizing to Sores providing a trade off between quality of the solution and speed. Furthermore, a fast but nonoptimal algorithm for generalizing to Chares is known. For each of the two language classes we give an efficient algorithm returning a minimal generalization from the given finite sample to an element of the fixed language class; such generalizations are called descriptive. In this sense, both our algorithms are optimal.

Croitoru, Cosmina; Kötzing, Timo A Normal Form for Argumentation Frameworks. Theorie and Applications of Formal Argumentation (TAFA) 2013: 3245
We study formal argumentation frameworks as introduced by Dung (1995). We show that any such argumentation framework can be syntactically augmented into a normal form (having a simplified attack relation), preserving the semantic properties of original arguments. An argumentation framework is in normal form if no argument attacks a conflicting pair of arguments. An augmentation of an argumentation framework is obtained by adding new arguments and changing the attack relation such that the acceptability status of original arguments is maintained in the new framework. Furthermore, we define joinnormal semantics leading to augmentations of the joined argumentation frameworks. Also, a rewriting technique which transforms in cubic time a given argumentation framework into a normal form is devised.

Bailly, Gilles; Oulasvirta, Antti; Kötzing, Timo; Hoppe, Sabrina MenuOptimizer: interactive optimization of menu systems. Symposium on User Interface Software and Technology (UIST) 2013: 331342
Menu systems are challenging to design because design spaces are immense, and several human factors affect user behavior. This paper contributes to the design of menus with the goal of interactively assisting designers with an optimizer in the loop. To reach this goal, 1) we extend a predictive model of user performance to account for expectations as to item groupings; 2) we adapt an ant colony optimizer that has been proven efficient for this class of problems; and 3) we present MenuOptimizer, a set of interactions integrated into a real interface design tool (QtDesigner). MenuOptimizer supports designers' abilities to cope with uncertainty and recognize good solutions. It allows designers to delegate combinatorial problems to the optimizer, which should solve them quickly enough without disrupting the design process. We show evidence that satisfactory menu designs can be produced for complex problems in minutes.

Case, John; Kötzing, Timo Learning secrets interactively. Dynamic modeling in inductive inference. Information and Computation 2012: 6073
Introduced is a new inductive inference paradigm, dynamic modeling. Within this learning paradigm, for example, function \(h\) learns function \(g\) iff, in the \(i\)th iteration, \(h\) and \(g\) both produce output, \(h\) gets the sequence of all outputs from \(g\) in prior iterations as input, \(g\) gets all the outputs from \(h\) in prior iterations as input, and, from some iteration on, the sequence of \(h\)'s outputs will be programs for the output sequence of \(g\). Dynamic modeling provides an idealization of, for example, a social interaction in which \(h\) seeks to discover program models of \(g\)'s behavior it sees in interacting with \(g\), and \(h\) openly discloses to \(g\) its sequence of candidate program models to see what \(g\) says back. Sample results: every \(g\) can be so learned by some \(h\); there are \(g\) that can only be learned by an \(h\) if \(g\) can also learn that \(h\) back; there are extremely secretive \(h\) which cannot be learned back by any \(g\) they learn, but which, nonetheless, succeed in learning infinitely many \(g\); quadratic time learnability is strictly more powerful than linear time learnability. This latter result, as well as others, follows immediately from general correspondence theorems obtained from a unified approach to the paradigms within inductive inference. Many proofs, some sophisticated, employ machine selfreference, a.k.a., recursion theorems.

Alcaraz, Nicolas; Friedrich, Tobias; Kötzing, Timo; Krohmer, Anton; Müller, Joachim; Pauling, Josch; Baumbach, Jan Efficient Key Pathway Mining: Combining Networks and OMICS Data. Integrative Biology 2012: 756764
Systems biology has emerged over the last decade. Driven by the advances in sophisticated measurement technology the research community generated huge molecular biology data sets. This comprises rather static data on the interplay of biological entities, for instance proteinprotein interaction network data, as well as quite dynamic data collected for studying the behavior of individual cells or tissues in accordance to changing environmental conditions, such as DNA microarrays or RNA sequencing. Here we bring the two different data types together in order to gain higher level knowledge. We introduce a significantly improved version of the KeyPathwayMiner software framework. Given a biological network modelled as graph and a set of expression studies, KeyPathwayMiner efficiently finds and visualizes connected subnetworks where most components are expressed in most cases. It finds all maximal connected subnetworks where all nodes but k exceptions are expressed in all experimental studies but at least l exceptions. We demonstrate the power of the new approach by comparing it to similar approaches with gene expression data previously used to study Huntington's disease. In addition, we demonstrate KeyPathwayMiner's flexibility and applicability to nonarray data by analyzing genomescale DNA methylation profiles from colorectal tumor cancer patients. KeyPathwayMiner release 2 is available as a Cytoscape plugin and online at http://keypathwayminer.mpiinf.mpg.de.

Case, John; Kötzing, Timo ComputabilityTheoretic Learning Complexity. Philosophical Transactions of the Royal Society A 2012: 35703596
Initially discussed are some of Alan Turing's wonderfully profound and influential ideas about mind and mechanismincluding regarding their connection to the main topic of the present study, which is within the field of computabilitytheoretic learning theory. Herein is investigated the part of this field concerned with the algorithmic, trialanderror inference of eventually correct programs for functions from their data points. As to the main content of this study: in prior papers, beginning with the seminal work by Freivalds et al. in 1995, the notion of intrinsic complexity is used to analyse the learning complexity of sets of functions in a Goldstyle learning setting. Herein are pointed out some weaknesses of this notion. Offered is an alternative based on epitomizing sets of functionssets that are learnable under a given learning criterion, but not under other criteria that are not at least as powerful. To capture the idea of epitomizing sets, new reducibility notions are given based on robust learning (closure of learning under certain sets of computable operators). Various degrees of epitomizing sets are characterized as the sets complete with respect to corresponding reducibility notions! These characterizations also provide an easy method for showing sets to be epitomizers, and they are then employed to prove several sets to be epitomizing. Furthermore, a scheme is provided to generate easily very strong epitomizers for a multitude of learning criteria. These strong epitomizers are the socalled selflearning sets, previously applied by Case & Kötzing in 2010. These strong epitomizers can be easily generated and employed in a myriad of settings to witness with certainty the strict separation in learning power between the criteria so epitomized and other not as powerful criteria!

Kötzing, Timo; Neumann, Frank; Röglin, Heiko; Witt, Carsten Theoretical analysis of two ACO approaches for the traveling salesman problem. Swarm Intelligence 2012: 121
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.

Heinz, Jeffrey; Kasprzik, Anna; Kötzing, Timo Learning in the limit with latticestructured hypothesis spaces. Theoretical Computer Science 2012: 111127
We define a collection of language classes which are TxtExlearnable (learnable in the limit from positive data). The learners map any data input to an element of a fixed lattice, and keep the least upper bound of all lattice elements thus obtained as the current hypothesis. Each element of the lattice is a grammar for a language, and the learner climbs the lattice searching for the right element. We call these classes in our collection lattice classes. We provide several characterizations of lattice classes and their learners, which suggests they are very natural. In particular, we show that any class of languages is a lattice class iff it is TxtExlearnable consistently, conservatively, setdrivenly, and strongly monotonically. We show several language classes previously discussed in the literature to be lattice classes, including the locally ktestable classes, the piecewise ktestable classes, the kreversible languages and the pattern languages. We also show that lattice classes contain three previously known collections of language classes: string extension language classes, functiondistinguishable language classes, and closedset systems. Finally, the lattice perspective helps analyze the learning of these classes. Illustrations include querylearning results in dependence on the lattice structure, characterizations of closure properties and the VCdimension of lattice classes in terms of lattice properties.

Jain, Sanjay; Kötzing, Timo; Stephan, Frank Enlarging Learnable Classes. Algorithmic Learning Theory (ALT) 2012: 3650
An early result in inductive inference shows that the class of Exlearnable sets is not closed under unions. In this paper we are interested in the following question: For what classes of functions is the union with an arbitrary Exlearnable class again Exlearnable, either effectively (in an index for a learner of an Exlearnable class) or noneffectively? We show that the effective case and the noneffective case separate, and we give a sufficient criterion for the effective case. Furthermore, we extend our notions to considering unions with classes of single functions, as well as to other learning criteria, such as finite learning and behaviorally correct learning. Furthermore, we consider the possibility of (effectively) extending learners to learn (infinitely) more functions. It is known that all Exlearners learning a dense set of functions can be effectively extended to learn infinitely more. It was open whether the learners learning a nondense set of functions can be similarly extended. We show that this is not possible, but we give an alternative split of all possible learners into two sets such that, for each of the sets, all learners from that set can be effectively extended. We analyze similar concepts also for other learning criteria.

Doerr, Benjamin; Hota, Ashish; Kötzing, Timo Ants easily solve stochastic shortest path problems. Genetic and Evolutionary Computation Conference (GECCO) 2012: 1724
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.

Baumbach, Jan; Friedrich, Tobias; Kötzing, Timo; Krohmer, Anton; Müller, Joachim; Pauling, Josch Efficient Algorithms for Extracting Biological Key Pathways with Global Constraints. Genetic and Evolutionary Computation Conference (GECCO) 2012: 169176
The integrated analysis of data of different types and with various interdependencies is one of the major challenges in computational biology. Recently, we developed KeyPathwayMiner, a method that combines biological networks modeled as graphs with diseasespecific genetic expression data gained from a set of cases (patients, cell lines, tissues, etc.). We aimed for finding all maximal connected subgraphs where all nodes but K are expressed in all cases but at most L, i.e. key pathways. Thereby, we combined biological networks with OMICS data, instead of analyzing these data sets in isolation. Here we present an alternative approach that avoids a certain bias towards hub nodes: We now aim for extracting all maximal connected subnetworks where all but at most K nodes are expressed in all cases but in total (!) at most L, i.e. accumulated over all cases and all nodes in a solution. We call this strategy GLONE (global node exceptions); the previous problem we call INES (individual node exceptions). Since finding GLONEcomponents is computationally hard, we developed an Ant Colony Optimization algorithm and implemented it with the KeyPathwayMiner Cytoscape framework as an alternative to the INES algorithms. KeyPathwayMiner 3.0 now offers both the INES and the GLONE algorithms. It is available as plugin from Cytoscape and online at http://keypathwayminer.mpiinf.mpg.de.

Kötzing, Timo; Sutton, Andrew M.; Neumann, Frank; O'Reilly, UnaMay The max problem revisited: the importance of mutation in genetic programming. Genetic and Evolutionary Computation Conference (GECCO) 2012: 13331340
This paper contributes to the rigorous understanding of genetic programming algorithms by providing runtime complexity analyses of the wellstudied Max problem. Several experimental studies have indicated that it is hard to solve the Max problem with crossoverbased algorithms. Our analyses show that different variants of the Max problem can provably be solved using simple mutationbased genetic programming algorithms. Our results advance the body of computational complexity analyses of genetic programming, indicate the importance of mutation in genetic programming, and reveal new insights into the behavior of mutationbased genetic programming algorithms.

Kötzing, Timo; Molter, Hendrik ACO Beats EA on a Dynamic PseudoBoolean Function. Parallel Problem Solving from Nature (PPSN) 2012: 113122
In this paper, we contribute to the understanding of the behavior of bioinspired 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.

Croitoru, Cosmina; Kötzing, Timo Deliberative Acceptability of Arguments. Starting AI Researcher Symposium (STAIRS) 2012: 7182
State of the art extension based argument acceptance is currently biased toward attacks: while the defending extension of an argument a is internally coherent, no such requirement is imposed on its attacking set. On the other hand, if we restrict ourselves only to conflictfree sets of attacking arguments, then we could have different attacking sets for a specified argument a (any two conflicting attackers of a must belong to different a's attacking sets). Having only one defending extension for all these attacking sets would contradict the deliberative nature of argumentation in the real world, where only the coherent sets of attacks matter and the defending sets of arguments depend on the former. In this paper we introduce a new type of acceptability of an argument, in which its attacking and defending sets of arguments are uniformly treated. We call it deliberative acceptance, discuss how this and the classical acceptance notions interrelate and analyze its computational properties. In particular, we prove that the corresponding decision problem is \(\Pi^P_2\)complete, but its restrictions on bipartite or cochordal argumentation frameworks are in \(P\).

Kötzing, Timo Iterative Learning from Positive Data and Counters. Algorithmic Learning Theory (ALT) 2011: 4054
We analyze iterative learning in the limit from positive data with the additional information provided by a counter. The simplest type of counter provides the current iteration number (counting up from 0 to infinity), which is known to improve learning power over plain iterative learning. We introduce five other (weaker) counter types, for example only providing some unbounded and nondecreasing sequence of numbers. Analyzing these types allows one to understand which properties of a counter learning can benefit from. For the iterative setting, we completely characterize the relative power of the learning criteria corresponding to the counter types. In particular, for our types, the only properties improving learning power are unboundedness and strict monotonicity. Furthermore, we show that each of our types of counter improves learning power over weaker ones in some settings; to this end, we analyze transductive and nonUshaped learning. Finally we show that, for iterative learning criteria with one of our types of counter, separations of learning criteria are necessarily witnessed by classes containing only infinite languages.

Doerr, Benjamin; Johannsen, Daniel; Kötzing, Timo; Lehre, Per Kristian; Wagner, Markus; Winzen, Carola Faster blackbox algorithms through higher arity operators. Foundations of Genetic Algorithms (FOGA) 2011: 163172
We extend the work of Lehre and Witt (GECCO 2010) on the unbiased blackbox model by considering higher arity variation operators. In particular, we show that already for binary operators the blackbox complexity of LeadingOnes drops from \(\Theta(n^2)\) for unary operators to \(O(n \log n)\). For OneMax, the \(\Omega(n \log n)\) unary blackbox complexity drops to \(O(n)\) in the binary case. For \(k\)ary operators, \(k \le n\), the OneMaxcomplexity further decreases to \(O(n / \log k)\).

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.

Doerr, Benjamin; Lengler, Johannes; Kötzing, Timo; Winzen, Carola Blackbox complexities of combinatorial problems. Genetic and Evolutionary Computation Conference (GECCO) 2011: 981988
Blackbox complexity is a complexity theoretic measure for how difficult a problem is to be optimized by a general purpose optimization algorithm. It is thus one of the few means trying to understand which problems are tractable for genetic algorithms and other randomized search heuristics. Most previous work on blackbox complexity is on artificial test functions. In this paper, we move a step forward and give a detailed analysis for the two combinatorial problems minimum spanning tree and singlesource shortest paths. Besides giving interesting bounds for their blackbox complexities, our work reveals that the choice of how to model the optimization problem is nontrivial here. This in particular comes true where the search space does not consist of bit strings and where a reasonable definition of unbiasedness has to be agreed on.

Kötzing, Timo; Sudholt, Dirk; Theile, Madeleine How crossover helps in pseudoboolean optimization. Genetic and Evolutionary Computation Conference (GECCO) 2011: 989996
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 crossoverbased GAs on the simple functions OneMax and Jump. First, we assess the potential speedup by crossover when combined with a fitnessinvariant 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.

Doerr, Benjamin; Kötzing, Timo; Winzen, Carola Too fast unbiased blackbox algorithms. Genetic and Evolutionary Computation Conference (GECCO) 2011: 20432050
Unbiased blackbox complexity was recently introduced as a refined complexity model for randomized search heuristics (Lehre and Witt, GECCO 2010). For several problems, this notion avoids the unrealistically low complexity results given by the classical model of Droste, Jansen, and Wegener (Theor. Comput. Sci. 2006). In this work, we show that for two natural problems the unbiased blackbox complexity remains artificially small. For the classical JumpK test function class and for a subclass of the wellknown Partition problem, we give mutationonly unbiased blackbox algorithms having complexity \(O(n \log n)\). Since the first problem usually needs \(\Theta(n^k)\) function evaluations to be optimized by standard heuristics and the second is even NPcomplete, these blackbox complexities seem not to indicate the true difficulty of the two problems for randomized search heuristics.

Kötzing, Timo; Neumann, Frank; Spöhel, Reto PAC learning and genetic programming. Genetic and Evolutionary Computation Conference (GECCO) 2011: 20912096
Genetic programming (GP) is a very successful type of learning algorithm that is hard to understand from a theoretical point of view. With this paper we contribute to the computational complexity analysis of genetic programming that has been started recently. We analyze GP in the wellknown PAC learning framework and point out how it can observe quality changes in the the evolution of functions by random sampling. This leads to computational complexity bounds for a linear GP algorithm for perfectly learning any member of a simple class of linear pseudoBoolean functions. Furthermore, we show that the same algorithm on the functions from the same class finds good approximations of the target function in less time.

Case, John; Kötzing, Timo Measuring Learning Complexity with Criteria Epitomizers. Symposium on Theoretical Aspects of Computer Science (STACS) 2011: 320331
In prior papers, beginning with the seminal work by Freivalds et al. 1995, the notion of intrinsic complexity is used to analyze the learning complexity of sets of functions in a Goldstyle learning setting. Herein are pointed out some weaknesses of this notion. Offered is an alternative based on epitomizing sets of functions  sets, which are learnable under a given learning criterion, but not under other criteria which are not at least as powerful. To capture the idea of epitomizing sets, new reducibility notions are given based on robust learning (closure of learning under certain classes of operators). Various degrees of epitomizing sets are characterized as the sets complete with respect to corresponding reducibility notions! These characterizations also provide an easy method for showing sets to be epitomizers, and they are, then, employed to prove several sets to be epitomizing. Furthermore, a scheme is provided to generate easily very strong epitomizers for a multitude of learning criteria. These strong epitomizers are socalled selflearning sets, previously applied by Case & Koetzing, 2010. These strong epitomizers can be generated and employed in a myriad of settings to witness the strict separation in learning power between the criteria so epitomized and other not as powerful criteria!

Case, John; Kötzing, Timo Solutions to Open Questions for NonUShaped Learning with Memory Limitations. Algorithmic Learning Theory (ALT) 2010: 285299
A Ushape occurs when a learner first learns, then unlearns, and, finally, relearns, some target concept. Within the framework of Inductive Inference, previous results have shown, for example, that Ushapes are unnecessary for explanatory learning, but are necessary for behaviorally correct learning. This paper solves the following two problems regarding nonUshaped learning posed in the prior literature. First, it is shown that there are classes learnable with three memory states that are not learnable nonUshapedly with any finite number of memory states. This result is surprising, as for learning with one or two memory states, Ushapes are known to be unnecessary. Secondly, it is shown that there is a class learnable memorylessly with a single feedback query such that this class is not learnable nonUshapedly memorylessly with any finite number of feedback queries. This result is complemented by the result that any class of infinite languages learnable memorylessly with finitely many feedback queries is so learnable without Ushapes  in fact, all classes of infinite languages learnable with complete memory are learnable memorylessly with finitely many feedback queries and without Ushapes. On the other hand, we show that there is a class of infinite languages learnable memorylessly with a single feedback query, which is not learnable without Ushapes by any particular bounded number of feedback queries.

Kötzing, Timo; Neumann, Frank; Röglin, Heiko; Witt, Carsten Theoretical Properties of Two ACO Approaches for the Traveling Salesman Problem. International Conference on Swarm Intelligence (ANTS) 2010: 324335
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.

Case, John; Kötzing, Timo Strongly NonUShaped Learning Results by General Techniques. Conference On Learning Theory (COLT) 2010: 181193
In learning, a semantic or behavioral Ushape occurs when a learner first learns, then unlearns, and, finally, relearns, some target concept (on the way to success). Within the framework of Inductive Inference, previous results have shown, for example, that such Ushapes are unnecessary for explanatory learning, but are necessary for behaviorally correct and nontrivial vacillatory learning. Herein we focus more on syntactic Ushapes. This paper introduces two general techniques and applies them especially to syntactic Ushapes in learning: one technique to show when they are necessary and one to show when they are unnecessary. The technique for the former is very general and applicable to a much wider range of learning criteria. It employs socalled selflearning classes of languages which are shown to characterize completely one criterion learning more than another. We apply these techniques to show that, for setdriven and partially setdriven learning, any kind of Ushapes are unnecessary. Furthermore, we show that Ushapes are not unnecessary in a strong way for iterative learning, contrasting an earlier result by Case and Moelius that semantic Ushapes are unnecessary for iterative learning.

Kötzing, Timo; Lehre, Per Kristian; Neumann, Frank; Oliveto, Pietro Simone Ant colony optimization and the minimum cut problem. Genetic and Evolutionary Computation Conference (GECCO) 2010: 13931400
Ant Colony Optimization (ACO) is a powerful metaheuristic for solving combinatorial optimization problems. With this paper we contribute to the theoretical understanding of this kind of algorithm by investigating the classical minimum cut problem. An ACO algorithm similar to the one that was proved successful for the minimum spanning tree problem is studied. Using rigorous runtime analyses we show how the ACO algorithm behaves similarly to Karger and Stein's algorithm for the minimum cut problem as long as the use of pheromone values is limited. Hence optimal solutions are obtained in expected polynomial time. On the other hand, we show that high use of pheromones has a negative effect, and the ACO algorithm may get trapped in local optima resulting in an exponential runtime to obtain an optimal solution. This result indicates that ACO algorithms may be inappropriate for finding minimum cuts.

Kasprzik, Anna; Kötzing, Timo String Extension Learning Using Lattices. Language and Automata Theory and Applications (LATA) 2010: 380391
The class of regular languages is not identifiable from positive data in Gold's language learning model. Many attempts have been made to define interesting classes that are learnable in this model, preferably with the associated learner having certain advantageous properties. Heinz '09 presents a set of language classes called String Extension (Learning) Classes, and shows it to have several desirable properties. In the present paper, we extend the notion of String Extension Classes by basing it on lattices and formally establish further useful properties resulting from this extension. Using lattices enables us to cover a larger range of language classes including the pattern languages, as well as to give various ways of characterizing String Extension Classes and its learners. We believe this paper to show that String Extension Classes are learnable in a very natural way, and thus worthy of further study.

Doerr, Benjamin; Johannsen, Daniel; Kötzing, Timo; Neumann, Frank; Theile, Madeleine More Effective Crossover Operators for the AllPairs Shortest Path Problem. Parallel Problem Solving from Nature (PPSN) 2010: 184193
The allpairs shortest path problem is the first nonartificial problem for which it was shown that adding crossover can significantly speed up a mutationonly 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 \(\Theta(n^3.25(\log n)^{0.25})\). 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 \(\emph{repair mechanisms}\) and \(\emph{parent selection}\). We show that repairing infeasible offspring leads to an improved expected optimization time of \(O(n^{3.2(\log n)^{0.2})\). As a second part of our study we prove that choosing parents that guarantee feasible offspring results in an even better optimization time of \(O(n^3 \log n)\). Both results show that already simple adjustments of the recombination operator can asymptotically improve the runtime of evolutionary algorithms.