Clean Citation Style 002
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Friedrich, Tobias; Rothenberger, Ralf Sharpness of the Satisfiability Threshold for NonUniform Random kSAT. International Joint Conference on Artificial Intelligence (IJCAI) 2019: 61516155
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; 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; 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.

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: 22722279
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.

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*.

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.

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; 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.

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.

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.

Rothenberger, Ralf; Grau, Sascha; Rossberg, Michael Dominating an stCut in a Network. Current Trends in Theory and Practice of Computer Science (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 \(V \setminus \{s,t\}\), we identify a \((\Delta + 1)\) approximation algorithm.

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.