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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. Its worst-case hardness lies at the core of computational complexity theory, for example in the form of NP-hardness and the (Strong) Exponential Time Hypothesis. In practice however, SAT instances can often be solved efficiently. This contradicting behavior has spawned interest in the average-case analysis of SAT and has triggered the development of sophisticated rigorous and non-rigorous 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, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a non-uniform distribution of the variables, which can result in distributions closer to industrial SAT instances.
We study satisfiability thresholds of non-uniform random 2-SAT with n variables and m clauses and with an arbitrary probability distribution (p_i)_{i in[n]} with p_1 >=slant p_2 >=slant ... >=slant 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 * (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].

Tobias Friedrich and Ralf Rothenberger. The Satisfiability Threshold for Non-Uniform Random 2-SAT. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 61:1-61:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{friedrich_et_al:LIPIcs.ICALP.2019.61, author = {Friedrich, Tobias and Rothenberger, Ralf}, title = {{The Satisfiability Threshold for Non-Uniform Random 2-SAT}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {61:1--61:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.61}, URN = {urn:nbn:de:0030-drops-106372}, doi = {10.4230/LIPIcs.ICALP.2019.61}, annote = {Keywords: random SAT, satisfiability threshold, sharpness, non-uniform distribution, 2-SAT} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous 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, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free 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 = (2k-1)/(k-1). This threshold is tight in the sense that instances with beta <= (2k-1)/(k-1)-epsilon for any constant epsilon > 0 are unsatisfiable with high probability (w.h.p.). For beta >= (2k-1)/(k-1)+epsilon, the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on beta.

Tobias Friedrich, Anton Krohmer, Ralf Rothenberger, Thomas Sauerwald, and Andrew M. Sutton. Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{friedrich_et_al:LIPIcs.ESA.2017.37, author = {Friedrich, Tobias and Krohmer, Anton and Rothenberger, Ralf and Sauerwald, Thomas and Sutton, Andrew M.}, title = {{Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {37:1--37:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.37}, URN = {urn:nbn:de:0030-drops-78356}, doi = {10.4230/LIPIcs.ESA.2017.37}, annote = {Keywords: satisfiability, random structures, random SAT, power law distribution, scale-freeness, phase transitions} }

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**Published in:** LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. In fact, the behavior of real-world networks and random graph models can be the complete opposite of one another, depending on the considered property. Brach, Cygan, Lacki, and Sankowski [SODA 2016] introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both deterministic properties and exploit them to design faster algorithms for a number of classical graph problems like transitive closure, maximum matching, determinant, PageRank, matrix inverse, counting triangles and maximum clique.
We complement the work of Brach et al. by showing that some well-studied random graph models exhibit both the mentioned PLB properties and additionally also a power-law lower bound on the degree distribution (PLB-L). All three properties hold with high probability for Chung-Lu 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 for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs.
In the second part of this work we study three classical NP-hard combinatorial optimization problems on PLB networks. It is known that on general graphs, a greedy algorithm, which chooses nodes in the order of their degree, only achieves an approximation factor of asymptotically at least logarithmic in the maximum degree for Minimum Vertex Cover and Minimum Dominating Set, and an approximation factor of asymptotically at least the maximum degree for Maximum Independent Set. We prove that the PLB-U property suffices such that the greedy approach achieves a constant-factor approximation for all three problems. We also show that all three combinatorial optimization problems are APX-complete, even if all PLB-properties hold. Hence, a PTAS cannot be expected, unless P=NP.

Ankit Chauhan, Tobias Friedrich, and Ralf Rothenberger. Greed is Good for Deterministic Scale-Free Networks. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chauhan_et_al:LIPIcs.FSTTCS.2016.33, author = {Chauhan, Ankit and Friedrich, Tobias and Rothenberger, Ralf}, title = {{Greed is Good for Deterministic Scale-Free Networks}}, booktitle = {36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-027-9}, ISSN = {1868-8969}, year = {2016}, volume = {65}, editor = {Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.33}, URN = {urn:nbn:de:0030-drops-68682}, doi = {10.4230/LIPIcs.FSTTCS.2016.33}, annote = {Keywords: random graphs, power-law degree distribution, scale-free networks, PLB networks, approximation algorithms, vertex cover, dominating set, independent s} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

An estimated 30% of urban traffic is caused by search for parking spots [Shoup, 2005]. Suggesting routes along highly probable parking spots could reduce traffic. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. 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 per-street 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.

Tobias Arndt, Danijar Hafner, Thomas Kellermeier, Simon Krogmann, Armin Razmjou, Martin S. Krejca, Ralf Rothenberger, and Tobias Friedrich. Probabilistic Routing for On-Street Parking Search. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arndt_et_al:LIPIcs.ESA.2016.6, author = {Arndt, Tobias and Hafner, Danijar and Kellermeier, Thomas and Krogmann, Simon and Razmjou, Armin and Krejca, Martin S. and Rothenberger, Ralf and Friedrich, Tobias}, title = {{Probabilistic Routing for On-Street Parking Search}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {6:1--6:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.6}, URN = {urn:nbn:de:0030-drops-63489}, doi = {10.4230/LIPIcs.ESA.2016.6}, annote = {Keywords: parking search, on-street parking, probabilistic routing, constrained optimization, dataset} }

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