
Arndt, Tobias; Hafner, Danijar; Kellermeier, Thomas; Krogmann, Simon; Razmjou, Armin; Krejca, Martin S.; Rothernberger, Ralf; Friedrich, Tobias Probabilistic Routing for OnStreet Parking Search. European Symposium on Algorithms (ESA) 2016: 6:16:13
An estimated 30% of urban traffic is caused by search for parking spots [10]. Traffic could be reduced by suggesting effective routes leading along potential parking spots. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NPcomplete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a perstreet basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.

Baum, Moritz; Bläsius, Thomas; Gemsa, Andreas; Rutter, Ignaz; Wegner, Franziska Scalable Exact Visualization of Isocontours in Road Networks via MinimumLink Paths. European Symposium on Algorithms (ESA) 2016: 7:17:18
Isocontours in road networks represent the area that is reachable from a source within a given resource limit. We study the problem of computing accurate isocontours in realistic, largescale networks. We propose isocontours represented by polygons with minimum number of segments that separate reachable and unreachable components of the network. Since the resulting problem is not known to be solvable in polynomial time, we introduce several heuristics that run in (almost) linear time and are simple enough to be implemented in practice. A key ingredient is a new practical lineartime algorithm for minimumlink paths in simple polygons. Experiments in a challenging realistic setting show excellent performance of our algorithms in practice, computing nearoptimal solutions in a few milliseconds on average, even for long ranges.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Hyperbolic Random Graphs: Separators and Treewidth. European Symposium on Algorithms (ESA) 2016: 15:115:16
When designing and analyzing algorithms, one can obtain better and more realistic results for practical instances by assuming a certain probability distribution on the input. The worstcase runtime is then replaced by the expected runtime or by bounds that hold with high probability (whp), i.e., with probability \(1 \neq O(1/n)\), on the random input. Hyperbolic random graphs can be used to model complex realworld networks as they share many important properties such as a small diameter, a large clustering coefficient, and a powerlaw degreedistribution. Divide and conquer is an important algorithmic design principle that works particularly well if the instance admits small separators. We show that hyperbolic random graphs in fact have comparatively small separators. More precisely, we show that a hyperbolic random graph can be expected to have a balanced separator hierarchy with separators of size \(O(\sqrt(n^(3\beta)))\), \(O(log n)\), and \(O(1)\) if \(2 < \beta < 3\), \(\beta = 3\) and \(3 < \beta\), respectively (\(\beta\) is the powerlaw exponent). We infer that these graphs have whp a treewidth of \(O(\sqrt(n^(3  \beta)))\), \(O(log^{2}n)\), and \(O(log n)\), respectively. For \(2 < \beta < 3\), this matches a known lower bound. For the more realistic (but harder to analyze) binomial model, we still prove a sublinear bound on the treewidth. To demonstrate the usefulness of our results, we apply them to obtain fast matching algorithms and an approximation scheme for Independent Set.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton; Laue, Sören Efficient Embedding of ScaleFree Graphs in the Hyperbolic Plane. European Symposium on Algorithms (ESA) 2016: 16:116:18
Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scalefree graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require a runtime of \(\Omega(n^{2})\). Our algorithm achieves quasilinear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics like Loglikelihood and greedy routing our algorithm discovers embeddings that are very close to the ground truth.

Chauhan, Ankit; Friedrich, Tobias; Rothenberger, Ralf Greed is Good for Deterministic ScaleFree Networks. Foundations of Software Technology and Theoretical Computer Science (FSTTCS) 2016
Large realworld networks typically follow a powerlaw degree distribution. To study such networks, numerous random graph models have been proposed. However, realworld networks are not drawn at random. Therefore, Brach, Cygan, Lacki, and Sankowski [SODA 2016] introduced two natural deterministic conditions: (1) a powerlaw upper bound on the degree distribution (PLBU) and (2) powerlaw neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLBN). They showed that many realworld networks satisfy both deterministic properties and exploit them to design faster algorithms for a number of classical graph problems. We complement the work of Brach et al. by showing that some wellstudied random graph models exhibit both the mentioned PLB properties and additionally also a powerlaw lower bound on the degree distribution (PLBL). All three properties hold with high probability for ChungLu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part of this work we study three classical NPhard combinatorial optimization problems on PLB networks. It is known that on general graphs with maximum degree ∆, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Ω(ln∆)approximation for Minimum Vertex Cover and Minimum Dominating Set, and a Ω(∆)approximation for Maximum Independent Set. We prove that the PLBU property suffices for the greedy approach to achieve a constantfactor approximation for all three problems. We also show that all three combinatorial optimization problems are APXcomplete even if all PLBproperties holds hence, PTAS cannot be expected unless P=NP.

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 (μ+1) GA using uniform crossover on the fitness function Jumpk. 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 μ, 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 μ: 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 (μ+1) GA compared to the (1+1) EA, which requires θ(n^k). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.

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.

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 Ω={0,1,…,r−1}n. More precisely, we regard a variety of problem classes requesting the componentwise minimization of the distance to an unknown target vector z∈Ω. 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 Θ(nrlogn). If we change each selected position by either +1 or −1 (random choice), the optimization time reduces to Θ(nr+nlogn). If we use a random mutation strength i∈{0,1,…,r−1}n with probability inversely proportional to i and change the selected position by either +i or −i (random choice), then the optimization time becomes Θ(nlog(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.

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 achiev able 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.

Friedrich, Tobias ScaleFree Networks, Hyperbolic Geometry and Efficient Algorithms. Mathematical Foundations of Computer Science (MFCS) 2016: 4:1  4:3
The node degrees of large realworld networks often follow a powerlaw distribution. Such scalefree networks can be social networks, internet topologies, the web graph, power grids, or many other networks from literally hundreds of domains. The talk will introduce several mathematical models of scalefree networks (e.g. preferential attachment graphs, ChungLu graphs, hyperbolic random graphs) and analyze some of their properties (e.g. diameter, average distance, clustering). We then present several algorithms and distributed processes on and for these network models (e.g. rumor spreading, load balancing, deanonymization, embedding) and discuss a number of open problems. The talk assumes no prior knowledge about scalefree networks, distributed computing or hyperbolic geometry.

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, λMMASIB, cGA, and (1,λ)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.; 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+δ, we require only O(log 1/δ + 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^{(1/2ε)}). 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.

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; 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.

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; 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 \emphmutational 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.

Gao, Wanru; Friedrich, Tobias; Neumann, Frank FixedParameter Single Objective Search Heuristics for Minimum Vertex Cover. Parallel Problem Solving From Nature (PPSN) 2016: 740750
We consider how wellknown branching approaches for the classical minimum vertex cover problem can be turned into randomized initialization strategies with provable performance guarantees and investigate them by experimental investigations. Furthermore, we show how these techniques can be built into local search components and analyze a basic local search variant that is similar to a stateoftheart approach called NuMVC. Our experimental results for the two local search approaches show that making use of more complex branching strategies in the local search component can lead to better results on various benchmark graphs.

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.