
Alam, Md. Jawaherul; Bläsius, Thomas; Rutter, Ignaz; Ueckerdt, Torsten; Wolff, Alexander Pixel and Voxel Representations of Graphs. Graph Drawing (GD) 2015: 472486
We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimumsize representations is NPcomplete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for \(k\)outerplanar graphs with \(n\) vertices, \(\Theta(kn)\) pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, \(\Theta(n^2)\) voxels are always sufficient and sometimes necessary for any \(n\)vertex graph. We improve this bound to \(\Theta(n cdot \tau)\) for graphs of treewidth \(\tau\) and to \(O((g+1)^2 n log^2 n)\) for graphs of genus \(g\). In particular, planar graphs admit representations with \(O(nlog^2 n)\) voxels.

Bläsius, Thomas; Lehmann, Sebastian; Rutter, Ignaz Orthogonal Graph Drawing with Inflexible Edges. Conference on Algorithms and Complexity (CIAC) 2015: 6173
We consider the problem of creating plane orthogonal drawings of 4planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge \(e\) a natural number \(flex(e)\), its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge \(e\) has at most \(flex(e)\) bends. It is known that FlexDraw is NPhard if \(flex(e)=0\) for every edge \(e\) [7]. On the other hand, FlexDraw can be solved efficiently if \(flex(e) \ge1\) [2] and is trivial if \(flex(e) ge 2\) [1] for every edge \(e\). To close the gap between the NPhardness for \(flex(e)=0\) and the efficient algorithm for \(flex(e) ge 1\), we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility 0). We show that for any \(epsilon > 0\) FlexDraw is NPcomplete for instances with \(O(n^\epsilon)\) inflexible edges with pairwise distance \(\Omega(n^{1\epsilon})\) (including the case where they induce a matching). On the other hand, we give an FPTalgorithm with running time \(O(2^k cdot n cdot T_flow(n))\), where \(T_flow(n)\) is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and \(k\) is the number of inflexible edges having at least one endpoint of degree 4.

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.

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 \(delta = 10^{25}\).

Chauhan, Ankit; Rao, B. V. Raghavendra Parameterized Analogues of Probabilistic Computation. Conference on Algorithms and Discrete Applied Mathematics (CALDAM) 2015: 181192
We study structural aspects of randomized parameterized computation. We introduce a new class W[P]PFPT as a natural parameterized analogue of PP. Our definition uses the machine based characterization of the parameterized complexity class W[P] obtained by Chen et.al [TCS 2005]. We translate most of the structural properties and characterizations of the class PP to the new class W[P]PFPT. We study a parameterization of the polynomial identity testing problem based on the degree of the polynomial computed by the arithmetic circuit. We obtain a parameterized analogue of the well known SchwartzZippel lemma [Schwartz, JACM 80 and Zippel, EUROSAM 79]. Additionally, we introduce a parameterized variant of permanent, and prove its #W[1] completeness.

CordLandwehr, Andreas; Lenzner, Pascal Network Creation Games: Think Global  Act Local. Mathematical Foundations of Computer Science (MFCS) 2015: 248260
We investigate a noncooperative gametheoretic model for the formation of communication networks by selfish agents. Each agent aims for a central position at minimum cost for creating edges. In particular, the general model (Fabrikant et al., PODC'03) became popular for studying the structure of the Internet or social networks. Despite its significance, locality in this game was first studied only recently (Bilo et al., SPAA'14), where a worst case locality model was presented, which came with a high efficiency loss in terms of quality of equilibria. Our main contribution is a new and more optimistic view on locality: agents are limited in their knowledge and actions to their local view ranges, but can probe different strategies and finally choose the best. We study the influence of our locality notion on the hardness of computing best responses, convergence to equilibria, and quality of equilibria. Moreover, we compare the strength of local versus nonlocal strategy changes. Our results address the gap between the original model and the worst case locality variant. On the bright side, our efficiency results are in line with observations from the original model, yet we have a nonconstant lower bound on the Price of Anarchy.

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

Doerr, Benjamin; Neumann, Frank; Sutton, Andrew M. Improved Runtime Bounds for the (1+1) EA on Random 3CNF Formulas Based on FitnessDistance Correlation. Genetic and Evolutionary Computation Conference (GECCO) 2015: 14151422
With this paper, we contribute to the theoretical understanding of randomized search heuristics by investigating their behavior on random 3SAT instances. We improve the results for the (1+1) EA obtained by Sutton and Neumann [PPSN 2014, 942951] in three ways. First, we reduce the upper bound by a linear factor and prove that the (1+1) EA obtains optimal solutions in time \(O(n log n)\) with high probability on asymptotically almost all highdensity satisfiable 3CNF formulas. Second, we extend the range of densities for which this bound holds to satisfiable formulas of at least logarithmic density. Finally, we complement these mathematical results with numerical experiments that summarize the behavior of the (1+1) EA on formulas along the density spectrum, and suggest that the implicit constants hidden in our bounds are low. Our proofs are based on analyzing the run of the algorithm by establishing a fitnessdistance correlation. This approach might be of independent interest and we are optimistic that it is useful for the analysis of randomized search heuristics in various other settings. To our knowledge, this is the first time that fitnessdistance correlation is explicitly used to rigorously prove a performance statement for an evolutionary algorithm.

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 \(Z^d\) 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 \(Z^d\). 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 \(\ell\).

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; 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.

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 \(\Omega(log n)\).

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.

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David Counting Homomorphisms to SquareFree Graphs, Modulo 2. International Colloquium on Automata, Languages, and Programming (ICALP) 2015: 642653
We study the problem \( oplus \)HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph \(H\). A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any \(H\) that contains no 4cycles, \(\oplus\)HomsToH is either in polynomial time or is \(\oplus\)Pcomplete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded treewidth. In particular, we focus on squarefree graphs, which are graphs without 4cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be treelike so that treelike decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

Kötzing, Timo; Lissovoi, Andrei; Witt, Carsten (1+1) EA on Generalized Dynamic OneMax. Foundations of Genetic Algorithms (FOGA) 2015: 4051
Evolutionary algorithms (EAs) perform well in settings involving uncertainty, including settings with stochastic or dynamic fitness functions. In this paper, we analyze the (1+1) EA on dynamically changing OneMax, as introduced by Droste (2003). We reprove the known results on first hitting times using the modern tool of drift analysis. We extend these results to search spaces which allow for more than two values per dimension. Furthermore, we make an anytime analysis as suggested by Jansen and Zarges (2014), analyzing how closely the (1+1) EA can track the dynamically moving optimum over time. We get tight bounds both for the case of bit strings, as well as for the case of more than two values per position. Surprisingly, in the latter setting, the expected quality of the search point maintained by the (1+1) EA does not depend on the number of values per dimension.

Rothenberger, Ralf; Grau, Sascha; Rossberg, Michael Dominating an stCut in a Network. Conference on Current Trends in Theory and Practice of Informatics (SOFSEM) 2015: 401411
We study an optimization problem with applications in design and analysis of resilient communication networks: given two vertices \(s, t\) in a graph \(G = (V,E)\), find a vertex set \(X subset V\) of minimum cardinality, such that \(X\) and its neighborhood constitute an \(st\) vertex separator. Although the problem naturally combines notions of graph connectivity and domination, its computational properties significantly differ from these relatives. In particular, we show that on general graphs the problem cannot be approximated to within a factor of \(2^{\log^{1\delta}n}\), with \(delta = 1 / \log\log^cn\) and arbitrary \(c<1/2\) (if P \(\neq\) NP). This inapproximability result even applies if the subgraph induced by a solution set has the additional constraint of being connected. Furthermore, we give a \(2\sqrt{n}\)approximation algorithm and study the problem on graphs with bounded node degree. With \(\Delta\) being the maximum degree of nodes \(V subseteq {s,t}\), we identify a \((Delta + 1)\) approximation algorithm.