Clean Citation Style 002
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Friedrich, Tobias; Krejca, Martin S.; Rothenberger, Ralf; Arndt, Tobias; Hafner, Danijar; Kellermeier, Thomas; Krogmann, Simon; Razmjou, Armin Routing for OnStreet Parking Search using Probabilistic Data. AI Communications 2019: 113124
A significant percentage of urban traffic is caused by the search for parking spots. One possible approach to improve this situation is to guide drivers along routes which are likely to have free parking spots. The task of finding such a route can be modeled as a probabilistic graph problem which is NPcomplete. Thus, we propose heuristic approaches for solving this problem and evaluate them experimentally. For this, we use probabilities of finding a parking spot, which are based on publicly available empirical data from TomTom International B.V. Additionally, we propose a heuristic that relies exclusively on conventional road attributes. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. Last, we complement our experiments with results from a field study, comparing the success rates of our algorithms against real human drivers.

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 Dynamic Uniform Constraints", we claimed a worstcase runtime of \(O(n D \log D)\) and \(O(n D)\) for the MultiObjective Evolutionary Algorithm and the MultiObjective \((\mu+(\lambda,\lambda))\) Genetic Algorithm, respectively, on linear profit functions under dynamic uniform constraint. The technique used to prove these results c contained an error. Instead, we correct this mistake and show a weaker bound of \(O(n D^2)\) for both algorithms.

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.

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
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.

Bilò, Davide; Friedrich, Tobias; Lenzner, Pascal; Melnichenko, Anna Geometric Network Creation Games. Symposium on Parallelism in Algorithms and Architectures (SPAA) 2019: 323332
Network Creation Games are a wellknown approach for explaining and analyzing the structure, quality and dynamics of realworld networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unitweight edges. In practice, e.g. when constructing a fiberoptic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edgeweighted graphs. We incorporate arbitrary edge weights by generalizing the wellknown model by Fabrikant et al.~[PODC'03] to edgeweighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the stateoftheart for the unitweight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight nonconstant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the nonmetric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the gametheoretic analogue of a variant of the classical Network Design Problem. Thus, lowcost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.

Friedrich, Tobias; Göbel, Andreas; Neumann, Frank; Quinzan, Francesco; Rothenberger, Ralf Greedy Maximization of Functions with Bounded Curvature Under Partition Matroid Constraints. Conference on Artificial Intelligence (AAAI) 2019
We investigate the performance of a deterministic GREEDY algorithm for the problem of maximizing functions under a partition matroid constraint. We consider nonmonotone submodular functions and monotone subadditive functions. Even though constrained maximization problems of monotone submodular functions have been extensively studied, little is known about greedy maximization of nonmonotone submodular functions or monotone subadditive functions. We give approximation guarantees for GREEDY on these problems, in terms of the curvature. We find that this simple heuristic yields a strong approximation guarantee on a broad class of functions. We discuss the applicability of our results to three realworld problems: Maximizing the determinant function of a positive semidefinite matrix, and related problems such as the maximum entropy sampling problem, the constrained maximum cut problem on directed graphs, and combinatorial auction games. We conclude that GREEDY is wellsuited to approach these problems. Overall, we present evidence to support the idea that, when dealing with constrained maximization problems with bounded curvature, one needs not search for (approximate) monotonicity to get good approximate solutions.

Roostapour, Vahid; Neumann, Aneta; Neumann, Frank; Friedrich, Tobias Pareto Optimization for Subset Selection with Dynamic Cost Constraints. Conference on Artificial Intelligence (AAAI) 2019
In this paper, we consider subset selection problems for functions \(f\) with constraints where the constraint bound \(B\) changes over time. We point out that adaptive variants of greedy approaches commonly used in the area of submodular optimization are not able to maintain their approximation quality. Investigating the recently introduced POMC Pareto optimization approach, we show that this algorithm efficiently computes a \( phi= (\alpha_f/2)(1\frac{1}{e^{\alpha_f}})\)approximation, where \(\alpha_f\) is the submodularity ratio, for each possible constraint bound \(b \leq B\). Furthermore, we show that POMC is able to adapt its set of solutions quickly in the case that \(B\) increases. Our experimental investigations for the influence maximization in social networks show the advantage of POMC over generalized greedy algorithms.

Bläsius, Thomas; Friedrich, Tobias; Lischeid, Julius; Meeks, Kitty; Schirneck, Martin Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling. Algorithm Engineering and Experiments (ALENEX) 2019: 130143
We devise an enumeration method for inclusionwise minimal hitting sets in hypergraphs. It has delay \(O(m^{k^\ast+1} \cdot n^2)\) and uses linear space. Hereby, \(n\) is the number of vertices, \(m\) the number of hyperedges, and \(k^\ast\) the rank of the transversal hypergraph. In particular, on classes of hypergraphs for which the cardinality \(k^\ast\) of the largest minimal hitting set is bounded, the delay is polynomial. The algorithm solves the extension problem for minimal hitting sets as a subroutine. We show that the extension problem is W[3]complete when parameterised by the cardinality of the set which is to be extended. For the subroutine, we give an algorithm that is optimal under the exponential time hypothesis. Despite these lower bounds, we provide empirical evidence showing that the enumeration outperforms the theoretical worstcase guarantee on hypergraphs arising in the profiling of relational databases, namely, in the detection of unique column combinations.

Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Meyer, Ulrich; Penschuck, Manuel; Weyand, Christopher Efficiently Generating Geometric Inhomogeneous and Hyperbolic Random Graphs. European Symposium on Algorithms (ESA) 2019
Hyperbolic random graphs (HRG) and geometric inhomogeneous random graphs (GIRG) are two similar generative network models that were designed to resemble complex real world networks. In particular, they have a powerlaw degree distribution with controllable exponent \(\beta\), and high clustering that can be controlled via the temperature \(T\). We present the first implementation of an efficient GIRG generator running in expected linear time. Besides varying temperatures, it also supports underlying geometries of higher dimensions. It is capable of generating graphs with ten million edges in under a second on commodity hardware. The algorithm can be adapted to HRGs. Our resulting implementation is the fastest sequential HRG generator, despite the fact that we support nonzero temperatures. Though nonzero temperatures are crucial for many applications, most existing generators are restricted to \(T = 0\). We also support parallelization, although this is not the focus of this paper. Moreover, we note that our generators draw from the correct probability distribution, i.e., they involve no approximation. Besides the generators themselves, we also provide an efficient algorithm to determine the nontrivial dependency between the average degree of the resulting graph and the input parameters of the GIRG model. This makes it possible to specify \(\bar{d}\) as input and obtain a graph with expected average degree \(\bar{d}\). Moreover, we investigate the differences between HRGs and GIRGs, shedding new light on the nature of the relation between the two models. Although HRGs represent, in a certain sense, a special case of the GIRG model, we find that a straightforward inclusion does not hold in practice. However, the difference is negligible for most use cases.

Friedrich, Tobias; Rothenberger, Ralf The Satisfiability Threshold for NonUniform Random 2SAT. International Colloquium on Automata, Languages and Programming (ICALP) 2019: 61:161:14
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worstcase hardness lies at the core of computational complexity theory, for example in the form of NPhardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the averagecase analysis of SAT and has triggered the development of sophisticated rigorous and nonrigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, most theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, realworld instances often exhibit large fluctuations in variable occurrence. This can be modeled by a nonuniform distribution of the variables, which can result in distributions closer to industrial SAT instances. We study satisfiability thresholds of nonuniform random 2SAT with n variables and m clauses and with an arbitrary probability distribution \((p_i)_{i \in [n]}\) with \(p_1 \geq p_2 \geq \dots \geq p_n > 0\) over the n variables. We show for \(p_1^2 = \Theta(\sum_{i=1^n p_i^2)\) that the asymptotic satisfiability threshold is at \(m = \Theta((1− \sum_{i=1^n p_i^2) / (p_1 cdot (\sum_{i=2^n p_i^2)^{1/2}))\) and that it is coarse. For \(p_1^2 = o( \sum_{i=1^n p_i^2)\) we show that there is a sharp satisfiability threshold at \(m = (\sum_{i=1^n p_i^2)^{−1}\). This result generalizes the seminal works by Chvatal and Reed [FOCS 1992] and by Goerdt [JCSS 1996].

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 \(10.5\) %.

Friedrich, Tobias; Rothenberger, Ralf Sharpness of the Satisfiability Threshold for NonUniform Random kSAT. International Joint Conference on Artificial Intelligence (IJCAI) 2019
Extended abstract.
We study nonuniform random kSAT on n variables with an arbitrary probability distribution p on the variable occurrences. The number \(t = t(n)\) of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We show that a threshold t(n) for random kSAT with an ensemble \((p_n)_{n\in\mathbb{N}}\) of arbitrary probability distributions on the variable occurrences is sharp if \(\p\_2^2 = O_n(t^{2/k})\) and \(\p\_infty = o_n(t^k/(2k1) \log^{(k1)/(2k1)(t))\). This result generalizes Friedgut’s sharpness result from uniform to nonuniform random kSAT and implies sharpness for thresholds of a wide range of random kSAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a powerlaw distribution.

Friedrich, Tobias From Graph Theory to Network Science: The Natural Emergence of Hyperbolicity. Symposium Theoretical Aspects of Computer Science (STACS) 2019: 5:1–5:9
Network science is driven by the question which properties large realworld networks have and how we can exploit them algorithmically. In the past few years, hyperbolic graphs have emerged as a very promising model for scalefree networks. The connection between hyperbolic geometry and complex networks gives insights in both directions: (1) Hyperbolic geometry forms the basis of a natural and explanatory model for realworld networks. Hyperbolic random graphs are obtained by choosing random points in the hyperbolic plane and connecting pairs of points that are geometrically close. The resulting networks share many structural properties for example with online social networks like Facebook or Twitter. They are thus well suited for algorithmic analyses in a more realistic setting. (2) Starting with a realworld network, hyperbolic geometry is wellsuited for metric embeddings. The vertices of a network can be mapped to points in this geometry, such that geometric distances are similar to graph distances. Such embeddings have a variety of algorithmic applications ranging from approximations based on efficient geometric algorithms to greedy routing solely using hyperbolic coordinates for navigation decisions.

Bläsius, Thomas; Friedrich, Tobias; Sutton, Andrew M. On the Empirical Time Complexity of ScaleFree 3SAT at the Phase Transition. Tools and Algorithms for the Construction and Analysis of Systems (TACAS) 2019: 117134
The hardness of formulas at the solubility phase transition of random propositional satisfiability (SAT) has been intensely studied for decades both empirically and theoretically. Solvers based on stochastic local search (SLS) appear to scale very well at the critical threshold, while complete backtracking solvers exhibit exponential scaling. On industrial SAT instances, this phenomenon is inverted: backtracking solvers can tackle large industrial problems, where SLSbased solvers appear to stall. Industrial instances exhibit sharply different structure than uniform random instances. Among many other properties, they are often heterogeneous in the sense that some variables appear in many while others appear in only few clauses. We conjecture that the heterogeneity of SAT formulas alone already contributes to the tradeoff in performance between SLS solvers and complete backtracking solvers. We empirically determine how the run time of SLS vs. backtracking solvers depends on the heterogeneity of the input, which is controlled by drawing variables according to a scalefree distribution. Our experiments reveal that the efficiency of complete solvers at the phase transition is strongly related to the heterogeneity of the degree distribution. We report results that suggest the depth of satisfying assignments in complete search trees is influenced by the level of heterogeneity as measured by a powerlaw exponent. We also find that incomplete SLS solvers, which scale well on uniform instances, are not affected by heterogeneity. The main contribution of this paper utilizes the scalefree random 3SAT model to isolate heterogeneity as an important factor in the scaling discrepancy between complete and SLS solvers at the uniform phase transition found in previous works.

Bläsius, Thomas; Fischbeck, Philipp; Friedrich, Tobias; Schirneck, Martin Understanding the Effectiveness of Data Reduction in Public Transportation Networks. Workshop on Algorithms and Models for the Web Graph (WAW) 2019: 87101
Given a public transportation network of stations and connections, we want to find a minimum subset of stations such that each connection runs through a selected station. Although this problem is NPhard in general, realworld instances are regularly solved almost completely by a set of simple reduction rules. To explain this behavior, we view transportation networks as hitting set instances and identify two characteristic properties, locality and heterogeneity. We then devise a randomized model to generate hitting set instances with adjustable properties. While the heterogeneity does influence the effectiveness of the reduction rules, the generated instances show that locality is the significant factor. Beyond that, we prove that the effectiveness of the reduction rules is independent of the underlying graph structure. Finally, we show that high locality is also prevalent in instances from other domains, facilitating a fast computation of minimum hitting sets.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton Cliques in Hyperbolic Random Graphs. Algorithmica 2018: 23242344
Most complex real world networks display scalefree features. This characteristic motivated the study of numerous random graph models with a powerlaw degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Krioukov et al. and has shown theoretically and empirically to fulfill all typical properties of real world networks, including powerlaw degree distribution and high clustering. We study cliques in hyperbolic random graphs \(G\) and present new results on the expected number of \(k\)cliques \(E[K_k]\) and the size of the largest clique \(\omega(G)\). We observe that there is a phase transition at powerlaw exponent \(\beta = 3\). More precisely, for \(\beta\)\(\in\)\((2,3)\) we prove \(E[K_k] = \) \(n^{k(3\beta)/2} \Theta(k)^{k}\) and \(\omega(G) = \) \(\Theta\)\((n^{(3\beta)/2})\), while for \(\beta \geq 3\) we prove \(E[K_k]=n \Theta(k)^{k}\) and \(\omega(G)=\Theta(\log(n)/ \log\log n)\). Furthermore, we show that for \(\beta \geq 3\), cliques in hyperbolic random graphs can be computed in time \(O(n)\). If the underlying geometry is known, cliques can be found with worstcase runtime \(O(m n^{2.5})\) for all values of \(\beta\).

Bringmann, Karl; Friedrich, Tobias; Krohmer, Anton Deanonymization of Heterogeneous Random Graphs in Quasilinear Time. Algorithmica 2018: 3397–3427
There are hundreds of online social networks with altogether billions of users. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy  it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial powerlaw network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an nvertex graph, our algorithm uses \(n^\varepsilon\) seed nodes (for an arbitrarily small \(\varepsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1+\Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.

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.

Bläsius, Thomas; Friedrich, Tobias; Krohmer, Anton; Laue, Sören Efficient Embedding of ScaleFree Graphs in the Hyperbolic Plane. IEEE/ACM Transactions on Networking 2018: 920933
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 at least a quadratic runtime. 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.

Friedrich, Tobias; Krohmer, Anton On the diameter of hyperbolic random graphs. SIAM Journal on Discrete Mathematics 2018: 13141334
Large realworld networks are typically scalefree. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing is close to optimal (Boguñá, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation has pushed the interest in hyperbolic networks as a natural model for scalefree networks. Hyperbolic random graphs follow a power law degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573–585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is \(O(\)\((\log n)\)\(^{32/((3  \beta)(5  \beta))+1})\)(Kiwi and Mitsche. ANALCO, pp. 26–39, 2015). We present two much simpler proofs for an improved upper bound of \(O((\log n)\)\(^{2/(3  \beta)})\) and a lower bound of \(\Omega(\log n)\). If \(\beta > 3\), we show that the latter bound is tight by proving an upper bound of \(O(\log n)\) for the diameter.

Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Unbounded Discrepancy of Deterministic Random Walks on Grids. SIAM Journal on Discrete Mathematics 2018: 24412452
Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called rotorrouter model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave remarkably similar: Starting with the same initial configuration, the number of tokens in the rotorrouter model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer (2006) showed that on \(\mathbb{Z}^{d}\) the single vertex discrepancy is only a constant \(c_d\). Other authors also determined the precise value of \(c_d\) for \(d=1,2\). All these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph \(\mathbb{Z}^{d}\). We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions \(d\geq1\) and arbitrary discrepancies~\(\ell \geq 0\), we construct configurations that reach a discrepancy of at least \(\ell\).

Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Hyperbolic Embeddings for NearOptimal Greedy Routing. Algorithm Engineering and Experiments (ALENEX) 2018: 199208
Greedy routing computes paths between nodes in a network by successively moving to the neighbor closest to the target with respect to coordinates given by an embedding into some metric space. Its advantage is that only local information is used for routing decisions. We present different algorithms for generating graph embeddings into the hyperbolic plane that are well suited for greedy routing. In particular our embeddings guarantee that greedy routing always succeeds in reaching the target and we try to minimize the lengths of the resulting greedy paths. We evaluate our algorithm on multiple generated and real wold networks. For networks that are generally assumed to have a hidden underlying hyperbolic geometry, such as the Internet graph, we achieve nearoptimal results, i.e., the resulting greedy paths are only slightly longer than the corresponding shortest paths. In the case of the Internet graph, they are only \(6\%\) longer when using our best algorithm, which greatly improves upon the previous best known embedding, whose creation required substantial manual intervention.

Friedrich, Tobias; Quinzan, Francesco; Wagner, Markus Escaping Large Deceptive Basins of Attraction with Heavy Mutation Operators. Genetic and Evolutionary Computation Conference (GECCO) 2018: 293300
In many Evolutionary Algorithms (EAs), a parameter that needs to be tuned is that of the mutation rate, which determines the probability for each decision variable to be mutated. Typically, this rate is set to 1/n for the duration of the optimization, where n is the number of decision variables. This setting has the appeal that the expected number of mutated variables per iteration is one. In a recent theoretical study, it was proposed to sample the number of mutated variables from a powerlaw distribution. This results into a significantly higher probability on larger numbers of mutations, so that escaping local optima becomes more probable. In this paper, we propose another class of nonuniform mutation rates. We study the benefits of this operator in terms of averagecase blackbox complexity analysis and experimental comparison. We consider both pseudoBoolean artificial landscapes and combinatorial problems (the Minimum Vertex Cover and the Maximum Cut). We observe that our nonuniform mutation rates significantly outperform the standard choices, when dealing with landscapes that exhibit large deceptive basins of attraction.

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 \(\epsilon > 4/n\).

Gao, Wanru; Friedrich, Tobias; Neumann, Frank; Hercher, Christian Randomized Greedy Algorithms for Covering Problems. Genetic and Evolutionary Computation Conference (GECCO) 2018: 309315
Greedy algorithms provide a fast and often also effective solution to many combinatorial optimization problems. However, it is well known that they sometimes lead to low quality solutions on certain instances. In this paper, we explore the use of randomness in greedy algorithms for the minimum vertex cover and dominating set problem and compare the resulting performance against their deterministic counterpart. Our algorithms are based on a parameter \(\gamma\) which allows to explore the spectrum between uniform and deterministic greedy selection in the steps of the algorithm and our theoretical and experimental investigations point out the benefits of incorporating randomness into greedy algorithms for the two considered combinatorial optimization problems.

Bläsius, Thomas; Freiberger, Cedric; Friedrich, Tobias; Katzmann, Maximilian; MontenegroRetana, Felix; Thieffry, Marianne Efficient Shortest Paths in ScaleFree Networks with Underlying Hyperbolic Geometry. International Colloquium on Automata, Languages, and Programming (ICALP) 2018: 20:120:14
A common way to accelerate shortest path algorithms on graphs is the use of a bidirectional search, which simultaneously explores the graph from the start and the destination. It has been observed recently that this strategy performs particularly well on scalefree realworld networks. Such networks typically have a heterogeneous degree distribution (e.g., a powerlaw distribution) and high clustering (i.e., vertices with a common neighbor are likely to be connected themselves). These two properties can be obtained by assuming an underlying hyperbolic geometry. To explain the observed behavior of the bidirectional search, we analyze its running time on hyperbolic random graphs and prove that it is \(\tilde{O}(n\)\(^{2  1/ \alpha}+\)\(n^{1/(2\alpha)}\)\(+ \delta_{\max})\) with high probability, where \(\alpha\)\(\in\)\((0.5, 1)\) controls the powerlaw exponent of the degree distribution, and \(\delta_{\max}\) is the maximum degree. This bound is sublinear, improving the obvious worstcase linear bound. Although our analysis depends on the underlying geometry, the algorithm itself is oblivious to it.

Friedrich, Tobias; Göbel, Andreas; Quinzan, Francesco; Wagner, Markus Heavytailed Mutation Operators in SingleObjective Combinatorial Optimization. Parallel Problem Solving From Nature (PPSN) 2018: 134145
A core feature of evolutionary algorithms is their mutation operator. Recently, much attention has been devoted to the study of mutation operators with dynamic and nonuniform mutation rates. Following up on this line of work, we propose a new mutation operator and analyze its performance on the (1+1) Evolutionary Algorithm (EA). Our analyses show that this mutation operator competes with preexisting ones, when used by the (1+1)EA on classes of problems for which results on the other mutation operators are available. We present a jump function for which the performance of the (1+1)EA using any static uniform mutation and any restart strategy can be worse than the performance of the (1+1)EA using our mutation operator with no restarts. We show that the (1+1)EA using our mutation operator finds a (1/3)approximation ratio on any nonnegative submodular function in polynomial time. This performance matches that of combinatorial local search algorithms specifically designed to solve this problem. Finally, we evaluate experimentally the performance of the (1+1)EA using our operator, on realworld graphs of different origins with up to \(\sim\)37,000 vertices and \(\sim\)1.6 million edges. In comparison with uniform mutation and a recently proposed dynamic scheme our operator comes out on top on these instances.

Friedrich, Tobias; Rothenberger, Ralf Sharpness of the Satisfiability Threshold for NonUniform Random kSAT. Theory and Applications of Satisfiability Testing (SAT) 2018: 273291
Best Paper Award
We study nonuniform random kSAT on n variables with an arbitrary probability distribution p on the variable occurrences. The number \(t = t(n)\) of randomly drawn clauses at which random formulas go from asymptotically almost surely (a.a.s.) satisfiable to a.a.s. unsatisfiable is called the satisfiability threshold. Such a threshold is called sharp if it approaches a step function as n increases. We show that a threshold t(n) for random kSAT with an ensemble \((p_n)_{n\in\mathbb{N}}\) of arbitrary probability distributions on the variable occurrences is sharp if \(\p\_2^2 = O_n(t^{2/k})\) and \(\p\_infty = o_n(t^k/(2k1) \log^{(k1)/(2k1)(t))\). This result generalizes Friedgut’s sharpness result from uniform to nonuniform random kSAT and implies sharpness for thresholds of a wide range of random kSAT models with heterogeneous probability distributions, for example such models where the variable probabilities follow a powerlaw distribution.

Bläsius, Thomas; Eube, Jan; Feldtkeller, Thomas; Friedrich, Tobias; Krejca, Martin S.; Lagodzinski, J. A. Gregor; Rothenberger, Ralf; Severin, Julius; Sommer, Fabian; Trautmann, Justin Memoryrestricted Routing With Tiled Map Data. IEEE International Conference on Systems, Man, and Cybernetics (SMC) 2018: 33473354
Modern routing algorithms reduce query time by depending heavily on preprocessed data. The recently developed Navigation Data Standard (NDS) enforces a separation between algorithms and map data, rendering preprocessing inapplicable. Furthermore, map data is partitioned into tiles with respect to their geographic coordinates. With the limited memory found in portable devices, the number of tiles loaded becomes the major factor for run time. We study routing under these restrictions and present new algorithms as well as empirical evaluations. Our results show that, on average, the most efficient algorithm presented uses more than 20 times fewer tile loads than a normal A*.

Bläsius, Thomas; Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton; Striebel, Jonathan Towards a Systematic Evaluation of Generative Network Models. Workshop on Algorithms and Models for the Web Graph (WAW) 2018: 99114
Generative graph models play an important role in network science. Unlike realworld networks, they are accessible for mathematical analysis and the number of available networks is not limited. The explanatory power of results on generative models, however, heavily depends on how realistic they are. We present a framework that allows for a systematic evaluation of generative network models. It is based on the question whether realworld networks can be distinguished from generated graphs with respect to certain graph parameters. As a proof of concept, we apply our framework to four popular random graph models (ErdősRényi, BarabásiAlbert, ChungLu, and hyperbolic random graphs). Our experiments for example show that all four models are bad representations for Facebook's social networks, while ChungLu and hyperbolic random graphs are good representations for other networks, with different strengths and weaknesses.

Anand, S.; Bringmann, Karl; Friedrich, Tobias; Garg, Naveen; Kumar, Amit Minimizing Maximum (Weighted) FlowTime on Related and Unrelated Machines. Algorithmica 2017: 515536
In this paper we initiate the study of job scheduling on related and unrelated machines so as to minimize the maximum flow time or the maximum weighted flow time (when each job has an associated weight). Previous work for these metrics considered only the setting of parallel machines, while previous work for scheduling on unrelated machines only considered \(L_p, p < \infty\) norms. Our main results are: (1) we give an \(O(\epsilon^{3})\)competitive algorithm to minimize maximum weighted flow time on related machines where we assume that the machines of the online algorithm can process \(1+\epsilon\) units of a job in 1 timeunit (\(\epsilon\) speed augmentation). (2) For the objective of minimizing maximum flow time on unrelated machines we give a simple \(2/\epsilon\)competitive algorithm when we augment the speed by \(\epsilon\). For \(m\) machines we show a lower bound of \(\Omega(m)\) on the competitive ratio if speed augmentation is not permitted. Our algorithm does not assign jobs to machines as soon as they arrive. To justify this "drawback" we show a lower bound of \(\Omega(\log m)\) on the competitive ratio of immediate dispatch algorithms. In both these lower bound constructions we use jobs whose processing times are in \(\{1,\infty\}\), and hence they apply to the more restrictive subset parallel setting. (3) For the objective of minimizing maximum weighted flow time on unrelated machines we establish a lower bound of \(\Omega(\log m)\)on the competitive ratio of any online algorithm which is permitted to use \(s = O(1)\) speed machines. In our lower bound construction, job \(j\) has a processing time of \(p_j\) on a subset of machines and infinity on others and has a weight \(1/p_j\). Hence this lower bound applies to the subset parallel setting for the special case of minimizing maximum stretch.

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.

Friedrich, Tobias; Krohmer, Anton; Rothenberger, Ralf; Sutton, Andrew M. Phase Transitions for ScaleFree SAT Formulas. Conference on Artificial Intelligence (AAAI) 2017: 38933899
Recently, a number of nonuniform random satisfiability models have been proposed that are closer to practical satisfiability problems in some characteristics. In contrast to uniform random Boolean formulas, scalefree formulas have a variable occurrence distribution that follows a power law. It has been conjectured that such a distribution is a more accurate model for some industrial instances than the uniform random model. Though it seems that there is already an awareness of a threshold phenomenon in such models, there is still a complete picture lacking. In contrast to the uniform model, the critical density threshold does not lie at a single point, but instead exhibits a functional dependency on the powerlaw exponent. For scalefree formulas with clauses of length \(k = 2\), we give a lower bound on the phase transition threshold as a function of the scaling parameter. We also perform computational studies that suggest our bound is tight and investigate the critical density for formulas with higher clause lengths. Similar to the uniform model, on formulas with \(k \geq 3\), we find that the phase transition regime corresponds to a set of formulas that are difficult to solve by backtracking search.

Friedrich, Tobias; Neumann, Frank What's Hot in Evolutionary Computation. Conference on Artificial Intelligence (AAAI) 2017: 50645066
We provide a brief overview on some hot topics in the area of evolutionary computation. Our main focus is on recent developments in the areas of combinatorial optimization and realworld applications. Furthermore, we highlight recent progress on the theoretical understanding of evolutionary computing methods.

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.

Wagner, Markus; Friedrich, Tobias; Lindauer, Marius Improving local search in a minimum vertex cover solver for classes of networks. Congress on Evolutionary Computation (CEC) 2017: 17041711
For the minimum vertex cover problem, a wide range of solvers has been proposed over the years. Most classical exact approaches are encountering run time issues on massive graphs that are considered nowadays. A straightforward alternative approach is then to use heuristics, which make assumptions about the structure of the studied graphs. These assumptions are typically hardcoded and are hoped to work well for a wide range of networks  which is in conflict with the nature of broad benchmark sets. With this article, we contribute in two ways. First, we identify a component in an existing solver that influences its performance depending on the class of graphs, and we then customize instances of this solver for different classes of graphs. Second, we create the first algorithm portfolio for the minimum vertex cover to further improve the performance of a single integrated approach to the minimum vertex cover problem.

Friedrich, Tobias; Krohmer, Anton; Rothenberger, Ralf; Sauerwald, Thomas; Sutton, Andrew M. Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT. European Symposium on Algorithms (ESA) 2017: 37:137:15
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worstcase hardness of SAT lies at the core of computational complexity theory. The averagecase analysis of SAT has triggered the development of sophisticated rigorous and nonrigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, realworld instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scalefree distribution of the variables, which results in distributions closer to industrial SAT instances. We study random \(k\)SAT on \(n\) variables, \(m=\Theta(n)\) clauses, and a power law distribution on the variable occurrences with exponent \(\beta\). We observe a satisfiability threshold at \(\beta=(2k1)/(k1)\). This threshold is tight in the sense that instances with \(beta < (2k1)/(k1)\varepsilon\) for any constant \(\varepsilon>0\) are unsatisfiable with high probability (w.h.p.). For \(\beta\ge(2k1)/(k1)+\varepsilon\), the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clausevariable ratios \(m/n\); they are unsatisfiable above a ratio \(m/n\) that depends on \(\beta\).

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.

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

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.

Chauhan, Ankit; Friedrich, Tobias; Quinzan, Francesco Approximating Optimization Problems using EAs on ScaleFree Networks. Genetic and Evolutionary Computation Conference (GECCO) 2017: 235242
It has been experimentally observed that realworld networks follow certain topologicalproperties, such as smallworld, powerlaw etc. To study these networks, many random graph models, such as Preferential Attachment, have been proposed. In this paper, we consider the deterministic properties which capture powerlaw degree distribution and degeneracy. Networks with these properties are known as scalefree networks in the literature. Many interesting problems remain NPhard on scalefree networks. We study the relationship between scalefree properties and the approximationratio of some commonly used evolutionary algorithms. For the Vertex Cover, we observe experimentally that the \((1+1)\) EA always gives the better result than a greedy local search, even when it runs for only \(O(n, \log(n))\) steps. We give the construction of a scalefree network in which a multiobjective algorithm and a greedy algorithm obtain optimal solutions, while the \((1+1)\) EA obtains the worst possible solution with constant probability. We prove that for the Dominating Set, Vertex Cover, Connected Dominating Set and Independent Set, the \((1+1)\) EA obtains constantfactor approximation in expected run time \(O(n, \log(n))\) and \(O(n^4)\) respectively. Whereas, GSEMO gives even better approximation than \((1+1)\) EA in expected run time \(O(n^3)\) for Dominating Set, Vertex Cover and Connected Dominating Set on such networks.

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; 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.

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.

Friedrich, Tobias; Ihde, Sven; Keßler, Christoph; Lenzner, Pascal; Neubert, Stefan; Schumann, David Efficient Best Response Computation for Strategic Network Formation under Attack. Symposium on Algorithmic Game Theory (SAGT) 2017: 199211
Inspired by real world examples, e.g. the Internet, researchers have introduced an abundance of strategic games to study natural phenomena in networks. Unfortunately, almost all of these games have the conceptual drawback of being computationally intractable, i.e. computing a best response strategy or checking if an equilibrium is reached is NPhard. Thus, a main challenge in the field is to find tractable realistic network formation models. We address this challenge by investigating a very recently introduced model by Goyal et al. [WINE'16] which focuses on robust networks in the presence of a strong adversary who attacks (and kills) nodes in the network and lets this attack spread viruslike to neighboring nodes and their neighbors. Our main result is to establish that this natural model is one of the few exceptions which are both realistic and computationally tractable. In particular, we answer an open question of Goyal et al. by providing an efficient algorithm for computing a best response strategy, which implies that deciding whether the game has reached a Nash equilibrium can be done efficiently as well. Our algorithm essentially solves the problem of computing a minimal connection to a network which maximizes the reachability while hedging against severe attacks on the network infrastructure and may thus be of independent interest.

Friedrich, Tobias; Ihde, Sven; Keßler, Christoph; Lenzner, Pascal; Neubert, Stefan; Schumann, David Brief Announcement: Efficient Best Response Computation for Strategic Network Formation under Attack. Symposium on Parallelism in Algorithms and Architectures (SPAA) 2017: 321323
Inspired by real world examples, e.g. the Internet, researchers have introduced an abundance of strategic games to study natural phenomena in networks. Unfortunately, almost all of these games have the conceptual drawback of being computationally intractable, i.e. computing a best response strategy or checking if an equilibrium is reached is NPhard. Thus, a main challenge in the field is to find tractable realistic network formation models. We address this challenge by establishing that the recently introduced model by Goyal et al.[WINE'16], which focuses on robust networks in the presence of a strong adversary, is a rare exception which is both realistic and computationally tractable. In particular, we sketch an efficient algorithm for computing a best response strategy, which implies that deciding whether the game has reached a Nash equilibrium can be done efficiently as well. Our algorithm essentially solves the problem of computing a minimal connection to a network which maximizes the reachability while hedging against severe attacks on the network infrastructure.

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.

Arndt, Tobias; Hafner, Danijar; Kellermeier, Thomas; Krogmann, Simon; Razmjou, Armin; Krejca, Martin S.; Rothenberger, 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. 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.

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 \(1  O(1/n)\), 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
EATCS Best Paper Award
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: 33:133:15
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 \(\Delta\), a greedy algorithm, which chooses nodes in the order of their degree, only achieves a \(\Omega(\ln \Delta)\)approximation for Minimum Vertex Cover and Minimum Dominating Set, and a \(\Omega(\Delta)\)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.

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.

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 \(O(n\,\log n)\).

Bläsius, Thomas; Friedrich, Tobias; Schirneck, Martin The Parameterized Complexity of Dependency Detection in Relational Databases. International Symposium on Parameterized and Exact Computation (IPEC) 2016: 6:16:13
We study the parameterized complexity of classical problems that arise in the profiling of relational data. Namely, we characterize the complexity of detecting unique column combinations (candidate keys), functional dependencies, and inclusion dependencies with the solution size as parameter. While the discovery of uniques and functional dependencies, respectively, turns out to be W[2]complete, the detection of inclusion dependencies is one of the first natural problems proven to be complete for the class W[3]. As a side effect, our reductions give insights into the complexity of enumerating all minimal unique column combinations or functional dependencies.

Friedrich, Tobias ScaleFree Networks, Hyperbolic Geometry, and Efficient Algorithms. Symposium on Mathematical Foundations of Computer Science (MFCS) 2016: 4:14:3
Invited Talk
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.

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.

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.

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.

Friedrich, Tobias; Wagner, Markus Seeding the initial population of multiobjective evolutionary algorithms: A computational study. Applied Soft Computing 2015: 223230
Most experimental studies initialize the population of evolutionary algorithms with random genotypes. In practice, however, optimizers are typically seeded with good candidate solutions either previously known or created according to some problemspecific method. This seeding has been studied extensively for singleobjective problems. For multiobjective problems, however, very little literature is available on the approaches to seeding and their individual benefits and disadvantages. In this article, we are trying to narrow this gap via a comprehensive computational study on common realvalued test functions. We investigate the effect of two seeding techniques for five algorithms on 48 optimization problems with 2, 3, 4, 6, and 8 objectives. We observe that some functions (e.g., DTLZ4 and the LZ family) benefit significantly from seeding, while others (e.g., WFG) profit less. The advantage of seeding also depends on the examined algorithm.

Friedrich, Tobias; Krohmer, Anton Parameterized clique on inhomogeneous random graphs. Discrete Applied Mathematics 2015: 130138
Finding cliques in graphs is a classical problem which is in general NPhard and parameterized intractable. In typical applications like social networks or biological networks, however, the considered graphs are scalefree, i.e., their degree sequence follows a power law. Their specific structure can be algorithmically exploited and makes it possible to solve clique much more efficiently. We prove that on inhomogeneous random graphs with \(n\) nodes and power law exponent \(\beta\), cliques of size \(k\) can be found in time \(O(n)\) for \(\beta \ge 3\) and in time \(O(ne^{k^4})\) for \(2 < \beta < 3\).

Wagner, Markus; Bringmann, Karl; Friedrich, Tobias; Neumann, Frank Efficient optimization of many objectives by approximationguided evolution. European Journal of Operational Research 2015: 465479
Multiobjective optimization problems arise frequently in applications, but can often only be solved approximately by heuristic approaches. Evolutionary algorithms have been widely used to tackle multiobjective problems. These algorithms use different measures to ensure diversity in the objective space but are not guided by a formal notion of approximation. We present a framework for evolutionary multiobjective optimization that allows to work with a formal notion of approximation. This approximationguided evolutionary algorithm (AGE) has a worstcase runtime linear in the number of objectives and works with an archive that is an approximation of the nondominated objective vectors seen during the run of the algorithm. Our experimental results show that AGE finds competitive or better solutions not only regarding the achieved approximation, but also regarding the total hypervolume. For all considered test problems, even for many (i.e., more than ten) dimensions, AGE discovers a good approximation of the Pareto front. This is not the case for established algorithms such as NSGAII, SPEA2, and SMSEMOA. In this paper we compare AGE with two additional algorithms that use very fast hypervolumeapproximations to guide their search. This significantly speeds up the runtime of the hypervolumebased algorithms, which now allows a comparison of the underlying selection schemes.

Friedrich, Tobias; Neumann, Frank; Thyssen, Christian Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point. Evolutionary Computation 2015: 131159
Many optimization problems arising in applications have to consider several objective functions at the same time. Evolutionary algorithms seem to be a very natural choice for dealing with multiobjective problems as the population of such an algorithm can be used to represent the tradeoffs with respect to the given objective functions. In this paper, we contribute to the theoretical understanding of evolutionary algorithms for multiobjective problems. We consider indicatorbased algorithms whose goal is to maximize the hypervolume for a given problem by distributing \(\mu\) points on the Pareto front. To gain new theoretical insights into the behavior of hypervolumebased algorithms we compare their optimization goal to the goal of achieving an optimal multiplicative approximation ratio. Our studies are carried out for different Pareto front shapes of biobjective problems. For the class of linear fronts and a class of convex fronts, we prove that maximizing the hypervolume gives the best possible approximation ratio when assuming that the extreme points have to be included in both distributions of the points on the Pareto front. Furthermore, we investigate the choice of the reference point on the approximation behavior of hypervolumebased approaches and examine Pareto fronts of different shapes by numerical calculations.

Friedrich, Tobias; Neumann, Frank Maximizing Submodular Functions under Matroid Constraints by Evolutionary Algorithms. Evolutionary Computation 2015: 543558
Many combinatorial optimization problems have underlying goal functions that are submodular. The classical goal is to find a good solution for a given submodular function \(f\) under a given set of constraints. In this paper, we investigate the runtime of a simple single objective evolutionary algorithm called (1+1) EA and a multiobjective evolutionary algorithm called GSEMO until they have obtained a good approximation for submodular functions. For the case of monotone submodular functions and uniform cardinality constraints we show that the GSEMO achieves a \((11/e)\)approximation in expected polynomial time. For the case of monotone functions where the constraints are given by the intersection of \(k \ge 2\) matroids, we show that the (1+1) EA achieves a \((1 + k/\delta)\)approximation in expected polynomial time for any constant \(\delta > 0\). Turning to nonmonotone symmetric submodular functions with \(k \ge 1\) matroid intersection constraints, we show that the GSEMO achieves a \((1/((k+2)(1+\epsilon)))\)approximation in expected time \(O(n^{k+6 \log(n)/\epsilon)\).

Berenbrink, Petra; Cooper, Colin; Friedetzky, Tom; Friedrich, Tobias; Sauerwald, Thomas Randomized diffusion for indivisible loads. Journal of Computer and System Sciences 2015: 159185
We present a new randomized diffusionbased algorithm for balancing indivisible tasks (tokens) on a network. Our aim is to minimize the discrepancy between the maximum and minimum load. The algorithm works as follows. Every vertex distributes its tokens as evenly as possible among its neighbors and itself. If this is not possible without splitting some tokens, the vertex redistributes its excess tokens among all its neighbors randomly (without replacement). In this paper we prove several upper bounds on the load discrepancy for general networks. These bounds depend on some expansion properties of the network, that is, the second largest eigenvalue, and a novel measure which we refer to as refined local divergence. We then apply these general bounds to obtain results for some specific networks. For constantdegree expanders and torus graphs, these yield exponential improvements on the discrepancy bounds compared to the algorithm of Rabani, Sinclair, and Wanka. For hypercubes we obtain a polynomial improvement. In contrast to previous papers, our algorithm is vertexbased and not edgebased. This means excess tokens are assigned to vertices instead to edges, and the vertex reallocates all of its excess tokens by itself. This approach avoids nodes having "negative loads", but causes additional dependencies for the analysis.

Friedrich, Tobias; Hercher, Christian On the kernel size of clique cover reductions for random intersection graphs. Journal of Discrete Algorithms 2015: 128136
Covering all edges of a graph by a minimum number of cliques is a well known NP hard problem. For the parameter \(k\) being the maximal number of cliques to be used, the problem becomes fixed parameter tractable. However, assuming the Exponential Time Hypothesis, there is no kernel of subexponential size in the worstcase. We study the average kernel size for random intersection graphs with \(n\) vertices, edge probability \(p\), and clique covers of size \(k\). We consider the wellknown set of reduction rules of Gramm, Guo, Hüffner, and Niedermeier (2009) and show that with high probability they reduce the graph completely if \(p\) is bounded away from 1 and \(k < c \log n\) for some constant \(c > 0\) . This shows that for large probabilistic graph classes like random intersection graphs the expected kernel size can be substantially smaller than the known exponential worstcase bounds.

Paixão, Tiago; Badkobeh, Golnaz; Barton, Nick H.; Çörüş, Doğan; Dang, DucCuong; Friedrich, Tobias; Lehre, Per Kristian; Sudholt, Dirk; Sutton, Andrew; Trubenová, Barbora Toward a unifying framework for evolutionary processes. Journal of Theoretical Biology 2015: 2843
The theory of population genetics and evolutionary computation have been evolving separately for nearly 30 years. Many results have been independently obtained in both fields and many others are unique to its respective field. We aim to bridge this gap by developing a unifying framework for evolutionary processes that allows both evolutionary algorithms and population genetics models to be cast in the same formal framework. The framework we present here decomposes the evolutionary process into its several components in order to facilitate the identification of similarities between different models. In particular, we propose a classification of evolutionary operators based on the defining properties of the different components. We cast several commonly used operators from both fields into this common framework. Using this, we map different evolutionary and genetic algorithms to different evolutionary regimes and identify candidates with the most potential for the translation of results between the fields. This provides a unified description of evolutionary processes and represents a stepping stone towards new tools and results to both fields.

Friedrich, Tobias; He, Jun; Jansen, Thomas; Moraglio, Alberto Genetic and Evolutionary Computation. Theoretical Computer Science 2015: 12
Evolutionary computation and other natureinspired search heuristics are known and applied on a daily basis for 50 years. They remain an important tool in situations where difficult problems need to be solved and no good problemspecific solution is available. Thanks to continuous efforts directed at a theoretical foundation of this broad and complex set of heuristics we have a much improved understanding of their properties, strengths and limitations. This special issue contains a collection of theoretical analyses of quite different natureinspired heuristics, all presented in preliminary form either at the field’s largest conference, the Genetic and Evolutionary Computation Conference (GECCO 2013), or at a small and specialized workshop on Theory of Randomized Search Heuristics (ThRaSH 2013) that provides an opportunity for theoreticians in the field to meet and discuss their work since 2007. All articles presented here have been reworked and significantly expanded beyond their initial presentation and they all witness that theory is now developed well beyond the understanding of very simple evolutionary algorithms on simple example functions.

Fountoulakis, Nikolaos; Friedrich, Tobias; Hermelin, Danny On the averagecase complexity of parameterized clique. Theoretical Computer Science 2015: 1829
The \(k\)Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most wellknown NPcomplete and W[1]complete problems. Moreover, its averagecase complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the averagecase complexity of the \(k\)Clique problem on the parameter \(k\). To this end, we define two natural parameterized analogs of efficient averagecase algorithms. We then show that \(k\)Clique admits both analogues for ErdHo}sRényi random graphs of arbitrary density. We also show that \(k\)Clique is unlikely to admit either of these analogs for some specific computable input distribution.

Bringmann, Karl; Friedrich, Tobias; Klitzke, Patrick Efficient computation of twodimensional solution sets maximizing the epsilonindicator. Congress on Evolutionary Computation (CEC) 2015: 970977
The majority of empirical comparisons of multiobjective evolutionary algorithms (MOEAs) are performed on synthetic benchmark functions. One of the advantages of synthetic test functions is the apriori knowledge of the optimal Pareto front. This allows measuring the proximity to the optimal front for the solution sets returned by the different MOEAs. Such a comparison is only meaningful if the cardinality of all solution sets is bounded by some fixed \(k\). In order to compare MOEAs to the theoretical optimum achievable with \(k\) solutions, we determine best possible \(\epsilon\)indicator values achievable with solution sets of size \(k\), up to an error of \(\delta\). We present a new algorithm with runtime \(O(k cdot \log^2(\delta1))\), which is an exponential improvement regarding the dependence on the error \(\delta\) compared to all previous work. We show mathematical correctness of our algorithm and determine optimal solution sets for sets of cardinality \(k \in \2, 3, 4, 5, 10, 20, 50, 100, 1000\}\) for the well known test suits DTLZ, ZDT, WFG and LZ09 up to error \(\delta = 10^{25}\).

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.

Bringmann, Karl; Friedrich, Tobias; Hoefer, Martin; Rothenberger, Ralf; Sauerwald, Thomas UltraFast Load Balancing on ScaleFree Networks. International Colloquium on Automata, Languages and Programming (ICALP) 2015: 516527
The performance of large distributed systems crucially depends on efficiently balancing their load. This has motivated a large amount of theoretical research how an imbalanced load vector can be smoothed with local algorithms. For technical reasons, the vast majority of previous work focuses on regular (or almost regular) graphs including symmetric topologies such as grids and hypercubes, and ignores the fact that large networks are often highly heterogenous. We model large scalefree networks by ChungLu random graphs and analyze a simple local algorithm for iterative load balancing. On nnode graphs our distributed algorithm balances the load within \(O((\log~\log~n)^2)\) steps. It does not need to know the exponent \(beta in (2,3)\) of the powerlaw degree distribution or the weights \(w_i\) of the graph model. To the best of our knowledge, this is the first result which shows that loadbalancing can be done in doublelogarithmic time on realistic graph classes.

Friedrich, Tobias; Krohmer, Anton On the Diameter of Hyperbolic Random Graphs. International Colloquium on Automata, Languages and Programming (ICALP) 2015: 614625
Large realworld networks are typically scalefree. Recent research has shown that such graphs are described best in a geometric space. More precisely, the internet can be mapped to a hyperbolic space such that geometric greedy routing performs close to optimal (Boguna, Papadopoulos, and Krioukov. Nature Communications, 1:62, 2010). This observation pushed the interest in hyperbolic networks as a natural model for scalefree networks. Hyperbolic random graphs follow a powerlaw degree distribution with controllable exponent \(\beta\) and show high clustering (Gugelmann, Panagiotou, and Peter. ICALP, pp. 573585, 2012). For understanding the structure of the resulting graphs and for analyzing the behavior of network algorithms, the next question is bounding the size of the diameter. The only known explicit bound is \(O((\log n)\)\(^{32/((3\beta)(5\beta))})\) (Kiwi and Mitsche. ANALCO, pp. 2639, 2015). We present two much simpler proofs for an improved upper bound of \(O((\log n)\)\(^{2/(3\beta)})\) and a lower bound of \(\Omega(\log n)\). If the average degree is bounded from above by some constant, we show that the latter bound is tight by proving an upper bound of \(O(\log n)\).

Friedrich, Tobias; Krohmer, Anton Cliques in Hyperbolic Random Graphs. International Conference on Computer Communications (INFOCOM) 2015: 15441552
Most complex realworld networks display scalefree features. This motivated the study of numerous random graph models with a powerlaw degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Papadopoulos, Krioukov, Boguna, Vahdat (INFOCOM, pp. 29732981, 2010) and has shown theoretically and empirically to fulfill all typical properties of realworld networks, including powerlaw degree distribution and high clustering. We study cliques in hyperbolic random graphs \(G\) and present new results on the expected number of \(k\)cliques \(E[K_k]\) and the size of the largest clique \(\omega(G)\). We observe that there is a phase transition at powerlaw exponent \(\gamma = 3\). More precisely, for \(\gamma\) \(\in\) \((2,3)\) we prove \(E[K_k] = \)\(n^{k(3\gamma)/2}\)\(\Theta(k)^{k}\) and \(\omega(G) = \)\(\Theta(\)\(n^{(3\gamma)/2})\) while for \(\gamma \ge 3\) we prove \(E[K_k] = n \Theta(k)^{k}\) and \(\omega(G) = \Theta(\log(n)/\log \log n)\). We empirically compare the \(\omega(G)\) value of several scalefree random graph models with realworld networks. Our experiments show that the \(\omega(G)\)predictions by hyperbolic random graphs are much closer to the data than other scalefree random graph models.

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.

Friedrich, Tobias; Katzmann, Maximilian; Krohmer, Anton Unbounded Discrepancy of Deterministic Random Walks on Grids. International Symposium on Algorithms and Computation (ISAAC) 2015: 212222
Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs, called rotorrouter model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave remarkably similar: Starting with the same initial configuration, the number of tokens in the rotorrouter model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer (2006) showed that on \(\mathbb{Z}^d\) the single vertex discrepancy is only a constant \(c_d\). Other authors also determined the precise value of \(c_d\) for \(d=1,2\). All these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph \(\mathbb{Z}^d\). We show that this assumption is crucial by proving that otherwise the single vertex discrepancy can become arbitrarily large. For all dimensions \(d \ge 1\) and arbitrary discrepancies \(\ell \ge 0\), we construct configurations that reach a discrepancy of at least \(\ell\).

Bringmann, Karl; Friedrich, Tobias Convergence of HypervolumeBased Archiving Algorithms. IEEE Transactions on Evolutionary Computation 2014: 643657
Multiobjective evolutionary algorithms typically maintain a set of solutions. A crucial part of these algorithms is the archiving, which decides what solutions to keep. A \((\mu + \lambda)\) archiving algorithm defines how to choose in each generation \(\mu\) children from \(\mu\) parents and \(\lambda\) offspring together. We study mathematically the convergence behavior of hypervolumebased archiving algorithms. We distinguish two cases for the offspring generation. A bestcase view leads to a study of the effectiveness of archiving algorithms. It was known that all \((\mu + 1)\)archiving algorithms are ineffective, which means that a set with maximum hypervolume is not necessarily reached. We prove that for \(\lambda < \mu\), all archiving algorithms are ineffective. We also present upper and lower bounds for the achievable hypervolume for different classes of archiving algorithms. On the other hand, a worstcase view on the offspring generation leads to a study of the competitive ratio of archiving algorithms. This measures how much smaller hypervolumes are achieved due to not knowing the future offspring in advance. We present upper and lower bounds on the competitive ratio of different archiving algorithms and present an archiving algorithm, which is the first known computationally efficient archiving algorithm with constant competitive ratio.

Friedrich, Tobias; Rowe, Jonathan E. Genetic and Evolutionary Computation. Theoretical Computer Science 2014: 1

Doerr, Benjamin; Friedrich, Tobias; Sauerwald, Thomas Quasirandom Rumor Spreading. Transactions on Algorithms 2014: 9:19:35
We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks ("randomized rumor spreading"). In the classical model, in each round, each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within \(O(\log n)\) rounds on complete graphs, hypercubes, random regular graphs, ErdHo}sRényi random graphs, and Ramanujan graphs with probability \(1  o(1)\). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the abovementioned bounds still hold. In some cases, even better bounds than for the classical model can be shown.

Bringmann, Karl; Friedrich, Tobias; Krohmer, Anton Deanonymization of Heterogeneous Random Graphs in Quasilinear Time. European Symposium on Algorithms (ESA) 2014: 197208
There are hundreds of online social networks with billions of users in total. Many such networks publicly release structural information, with all personal information removed. Empirical studies have shown, however, that this provides a false sense of privacy  it is possible to identify almost all users that appear in two such anonymized network as long as a few initial mappings are known. We analyze this problem theoretically by reconciling two versions of an artificial powerlaw network arising from independent subsampling of vertices and edges. We present a new algorithm that identifies most vertices and makes no wrong identifications with high probability. The number of vertices matched is shown to be asymptotically optimal. For an \(n\)vertex graph, our algorithm uses \(n^\epsilon\) seed nodes (for an arbitrarily small \(\epsilon\)) and runs in quasilinear time. This improves previous theoretical results which need \(\Theta(n)\) seed nodes and have runtimes of order \(n^{1 + \Omega(1)}\). Additionally, the applicability of our algorithm is studied experimentally on different networks.

Bringmann, Karl; Friedrich, Tobias; Klitzke, Patrick Twodimensional subset selection for hypervolume and epsilonindicator. Genetic and Evolutionary Computation Conference (GECCO) 2014: 589596
The goal of biobjective optimization is to find a small set of good compromise solutions. A common problem for biobjective evolutionary algorithms is the following subset selection problem (SSP): Given \(n\) solutions \(P \subset \mathbb{R}^2\) in the objective space, select \(k\) solutions \(P^*\) from \(P\) that optimize an indicator function. In the hypervolume SSP we want to select \(k\) points \(P^*\) that maximize the hypervolume indicator \(I_{HYP}(P^*, r)\) for some reference point \(r \in R^2\). Similarly, the \(\epsilon\)indicator SSP aims at selecting \(\tilde k\) points \(P^*\) that minimize the \(\epsilon\)indicator \(I_{\epsilon}(P^*,\mathbb{R})\) for some reference set \(R \subset R^2\) of size \(m\) (which can be \(\mathbb{R}=P\)). We first present a new algorithm for the hypervolume SSP with runtime \(O(n (k + \log n))\). Our second main result is a new algorithm for the \(\epsilon\)indicator SSP with runtime \(O(n \log n + m \log m)\). Both results improve the current state of the art runtimes by a factor of (nearly) \(n\) and make the problems tractable for new applications. Preliminary experiments confirm that the theoretical results translate into substantial empirical runtime improvements.

Bringmann, Karl; Friedrich, Tobias; Klitzke, Patrick Generic Postprocessing via Subset Selection for Hypervolume and EpsilonIndicator. Parallel Problem Solving from Nature (PPSN) 2014: 518527
Most biobjective evolutionary algorithms maintain a population of fixed size \(\mu\) and return the final population at termination. During the optimization process many solutions are considered, but most are discarded. We present two generic postprocessing algorithms which utilize the archive of all nondominated solutions evaluated during the search. We choose the best \(\mu\) solutions from the archive such that the hypervolume or \(\epsilon\)indicator is maximized. This postprocessing costs no additional fitness function evaluations and has negligible runtime compared to most EMOAs. We experimentally examine our postprocessing for four standard algorithms (NSGAII, SPEA2, SMSEMOA, IBEA) on ten standard test functions (DTLZ 12,7, ZDT 13, WFG 36) and measure the average quality improvement. The median decrease of the distance to the optimal \(\epsilon\)indicator is \(95\%\), the median decrease of the distance to the optimal hypervolume value is \(86\%\). We observe similar performance on a realworld problem (wind turbine placement).

Friedrich, Tobias; Neumann, Frank Maximizing Submodular Functions under Matroid Constraints by Multiobjective Evolutionary Algorithms. Parallel Problem Solving from Nature (PPSN) 2014: 922931
Nominated for Best Paper Award
Many combinatorial optimization problems have underlying goal functions that are submodular. The classical goal is to find a good solution for a given submodular function f under a given set of constraints. In this paper, we investigate the runtime of a multiobjective evolutionary algorithm called GSEMO until it has obtained a good approximation for submodular functions. For the case of monotone submodular functions and uniform cardinality constraints we show that GSEMO achieves a \((1  1/e)\)approximation in expected time \(O(n^2(\log n+k))\), where \(k\) is the value of the given constraint. For the case of nonmonotone submodular functions with \(k\) matroid intersection constraints, we show that GSEMO achieves a \((1/(k + 2 + 1/k + \epsilon)\)approximation in expected time \(O(n^{k+5\log(n)/\epsilon)\).

Friedrich, Tobias; Sauerwald, Thomas; Stauffer, Alexandre Diameter and Broadcast Time of Random Geometric Graphs in Arbitrary Dimensions. Algorithmica 2013: 6588
A random geometric graph (RGG) is defined by placing \(n\) points uniformly at random in \([0,n^{1/d}]^d\), and joining two points by an edge whenever their Euclidean distance is at most some fixed \(r\). We assume that \(r\) is larger than the critical value for the emergence of a connected component with \(\Omega(n)\) nodes. We show that, with high probability (w.h.p.), for any two connected nodes with a Euclidean distance of \(\omega(\log n / r^{d1})\), their graph distance is only a constant factor larger than their Euclidean distance. This implies that the diameter of the largest connected component is \(\Theta(n^{1/d}/r)\) w.h.p. We also prove that the condition on the Euclidean distance above is essentially tight. We also analyze the following randomized broadcast algorithm on RGGs. At the beginning, only one node from the largest connected component of the RGG is informed. Then, in each round, each informed node chooses a neighbor independently and uniformly at random and informs it. We prove that w.h.p. this algorithm informs every node in the largest connected component of an RGG within \(\Theta(n^1/d/r+\log n)\) rounds.

Bringmann, Karl; Friedrich, Tobias; Igel, Christian; Voß, Thomas Speeding up manyobjective optimization by Monte Carlo approximations. Artificial Intelligence 2013: 2229
Many stateoftheart evolutionary vector optimization algorithms compute the contributing hypervolume for ranking candidate solutions. However, with an increasing number of objectives, calculating the volumes becomes intractable. Therefore, although hypervolumebased algorithms are often the method of choice for bicriteria optimization, they are regarded as not suitable for manyobjective optimization. Recently, Monte Carlo methods have been derived and analyzed for approximating the contributing hypervolume. Turning theory into practice, we employ these results in the ranking procedure of the multiobjective covariance matrix adaptation evolution strategy (MOCMAES) as an example of a stateoftheart method for vector optimization. It is empirically shown that the approximation does not impair the quality of the obtained solutions given a budget of objective function evaluations, while considerably reducing the computation time in the case of multiple objectives. These results are obtained on common benchmark functions as well as on two design optimization tasks. Thus, employing Monte Carlo approximations makes hypervolumebased algorithms applicable to manyobjective optimization.

Bringmann, Karl; Friedrich, Tobias Approximation quality of the hypervolume indicator. Artificial Intelligence 2013: 265290
In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space bounded from below by a reference point. Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator of sets of bounded size is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation ratio with the approximation ratio achieved by twodimensional sets maximizing the hypervolume indicator. We bound the optimal multiplicative approximation ratio of \(n\) points by \(1+\Theta(1/n)\) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of \(n\) points that maximize the hypervolume indicator. However, there is a provable gap between the two approximation ratios which is even exponential in the ratio between the largest and the smallest value of the front. We also examine the additive approximation ratio of the hypervolume indicator in two dimensions and prove that it achieves the optimal additive approximation ratio apart from a small ratio.

Friedrich, Tobias; Kroeger, Trent; Neumann, Frank Weighted preferences in evolutionary multiobjective optimization. Machine Learning and Cybernetics 2013: 139148
Evolutionary algorithms have been widely used to tackle multiobjective optimization problems. Incorporating preference information into the search of evolutionary algorithms for multiobjective optimization is of great importance as it allows one to focus on interesting regions in the objective space. Zitzler et al. have shown how to use a weight distribution function on the objective space to incorporate preference information into hypervolumebased algorithms. We show that this weighted information can easily be used in other popular EMO algorithms as well. Our results for NSGAII and SPEA2 show that this yields similar results to the hypervolume approach and requires less computational effort.

Friedrich, Tobias; Levine, Lionel Fast simulation of largescale growth models. Random Structures and Algorithms 2013: 185213
We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a "least action principle" which characterizes the odometer function of the growth process. Starting from an approximation for the odometer, we successively correct under and overestimates and provably arrive at the correct final state. The degree of speedup depends on the accuracy of the initial guess. Determining the size of the boundary fluctuations in growth models like internal diffusionlimited aggregation (IDLA) is a longstanding open problem in statistical physics. As an application of our method, we calculate the size of fluctuations over two orders of magnitude beyond previous simulations.

Vladislavleva, Ekaterina; Friedrich, Tobias; Neumann, Frank; Wagner, Markus Predicting the Energy Output of Wind Farms Based on Weather Data: Important Variables and their Correlation. Renewable Energy 2013: 236243
Wind energy plays an increasing role in the supply of energy world wide. The energy output of a wind farm is highly dependent on the weather conditions present at its site. If the output can be predicted more accurately, energy suppliers can coordinate the collaborative production of different energy sources more efficiently to avoid costly overproduction. In this paper, we take a computer science perspective on energy prediction based on weather data and analyze the important parameters as well as their correlation on the energy output. To deal with the interaction of the different parameters, we use symbolic regression based on the genetic programming tool DataModeler. Our studies are carried out on publicly available weather and energy data for a wind farm in Australia. We report on the correlation of the different variables for the energy output. The model obtained for energy prediction gives a very reliable prediction of the energy output for newly supplied weather data.

Fellows, Michael R.; Friedrich, Tobias; Hermelin, Danny; Narodytska, Nina; Rosamond, Frances A. Constraint satisfaction problems: Convexity makes AllDifferent constraints tractable. Theoretical Computer Science 2013: 8189
We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of realworld and combinatorial problems, like scheduling, frequency allocations, and graph coloring problems. As this problem is known to be NPcomplete, we investigate under which further assumptions it becomes tractable. We observe that a crucial property seems to be the convexity of the variable domains and constraints. Our main contribution is an extensive study of the complexity of Multiple AllDiff CSPs for a set of natural parameters, like maximum domain size and maximum size of the constraint scopes. We show that, depending on the parameter, convexity can make the problem tractable even though it is provably intractable in general. Interestingly, the convexity of constraints is the key property in achieving fixed parameter tractability, while the convexity of domains does not usually make the problem easier.

Wagner, Markus; Friedrich, Tobias Efficient parent selection for ApproximationGuided Evolutionary multiobjective optimization. Congress on Evolutionary Computation (CEC) 2013: 18461853
The Pareto front of a multiobjective optimization problem is typically very large and can only be approximated. ApproximationGuided Evolution (AGE) is a recently presented evolutionary multiobjective optimization algorithm that aims at minimizing iteratively the approximation factor, which measures how well the current population approximates the Pareto front. It outperforms stateoftheart algorithms for problems with many objectives. However, AGE's performance is not competitive on problems with very few objectives. We study the reason for this behavior and observe that AGE selects parents uniformly at random, which has a detrimental effect on its performance. We then investigate different algorithmspecific selection strategies for AGE. The main difficulty here is finding a computationally efficient selection scheme which does not harm AGEs linear runtime in the number of objectives. We present several improved selections schemes that are computationally efficient and substantially improve AGE on lowdimensional objective spaces, but have no negative effect in highdimensional objective spaces.

Bringmann, Karl; Friedrich, Tobias Parameterized averagecase complexity of the hypervolume indicator. Genetic and Evolutionary Computation Conference (GECCO) 2013: 575582
Nominated for Best Paper Award (EMO Track)
The hypervolume indicator (HYP) is a popular measure for the quality of a set of \(n\) solutions in \(\mathbb{R}^d\). We discuss its asymptotic worstcase runtimes and several lower bounds depending on different complexitytheoretic assumptions. Assuming that P \(\neq\) NP, there is no algorithm with runtime \(poly(n,d)\). Assuming the exponential time hypothesis, there is no algorithm with runtime \(n^{o(d)}\). In contrast to these worstcase lower bounds, we study the averagecase complexity of HYP for points distributed i.i.d. at random on a \(d\)dimensional simplex. We present a general framework which translates any algorithm for HYP with worstcase runtime \(n^{f(d)}\) to an algorithm with worstcase runtime \(n^{f(d) +1}\) and fixedparametertractable (FPT) averagecase runtime. This can be used to show that HYP can be solved in expected time \(O(d^{d^2/2 n + d n^2)\), which implies that HYP is FPT on average while it is W1hard in the worstcase. For constant dimension \(d\) this gives an algorithm for HYP with runtime \(O(n^2)\) on average. This is the first result proving that HYP is asymptotically easier in the average case. It gives a theoretical explanation why most HYP algorithms perform much better on average than their theoretical worstcase runtime predicts.

Anand, S.; Bringmann, Karl; Friedrich, Tobias; Garg, Naveen; Kumar, Amit Minimizing Maximum (Weighted) FlowTime on Related and Unrelated Machines. International Colloquium on Automata, Languages and Programming (ICALP) 2013: 1324
In this paper we initiate the study of job scheduling on related and unrelated machines so as to minimize the maximum flow time or the maximum weighted flow time (when each job has an associated weight). Previous work for these metrics considered only the setting of parallel machines, while previous work for scheduling on unrelated machines only considered \(L_p, p < \infty\) norms. Our main results are: (1) we give an \(O(\epsilon^{3})\)competitive algorithm to minimize maximum weighted flow time on related machines where we assume that the machines of the online algorithm can process \(1+\epsilon\) units of a job in 1 timeunit (\(\epsilon\) speed augmentation). (2) For the objective of minimizing maximum flow time on unrelated machines we give a simple \(2/\epsilon\)competitive algorithm when we augment the speed by \(\epsilon\). For \(m\) machines we show a lower bound of \(\Omega(m)\) on the competitive ratio if speed augmentation is not permitted. Our algorithm does not assign jobs to machines as soon as they arrive. To justify this "drawback" we show a lower bound of \(\Omega(\log m)\) on the competitive ratio of immediate dispatch algorithms. In both these lower bound constructions we use jobs whose processing times are in \(\{1,\infty\}\), and hence they apply to the more restrictive subset parallel setting. (3) For the objective of minimizing maximum weighted flow time on unrelated machines we establish a lower bound of \(\Omega(\log m)\)on the competitive ratio of any online algorithm which is permitted to use \(s = O(1)\) speed machines. In our lower bound construction, job j has a processing time of \(p_j\) on a subset of machines and infinity on others and has a weight \(1/p_j\). Hence this lower bound applies to the subset parallel setting for the special case of minimizing maximum stretch.

Bringmann, Karl; Friedrich, Tobias Exact and Efficient Generation of Geometric Random Variates and Random Graphs. International Colloquium on Automata, Languages, and Programming (ICALP) 2013: 267278