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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-r minors have constant density. More precisely, the formulas are ∃ x₁… x_k#y φ(x_1,…,x_k, y) > N, where φ is an FO-formula. If φ is quantifier-free, we can extend this result to nowhere dense graph classes with an almost linear FPT run time. Lifting this result further to slightly more general graph classes, namely almost nowhere dense classes, where the size of depth-r clique minors is subpolynomial, is impossible unless FPT = W[1]. On the other hand, in almost nowhere dense classes we can approximate such counting formulas with a small additive error. Note those counting formulas are contained in FOC({>}) but not FOC₁(𝐏).
In particular, it follows that partial covering problems, such as partial dominating set, have fixed parameter algorithms on nowhere dense graph classes with almost linear running time.

Jan Dreier, Daniel Mock, and Peter Rossmanith. Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 43:1-43:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dreier_et_al:LIPIcs.ESA.2023.43, author = {Dreier, Jan and Mock, Daniel and Rossmanith, Peter}, title = {{Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {43:1--43:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.43}, URN = {urn:nbn:de:0030-drops-186961}, doi = {10.4230/LIPIcs.ESA.2023.43}, annote = {Keywords: nowhere dense, sparsity, counting logic, dominating set, FPT} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth.
Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class C of graphs is flip-flat if for every fixed radius r, every sufficiently large set of vertices of a graph G ∈ C contains a large subset of vertices with mutual distance larger than r, where the distance is measured in some graph G' that can be obtained from G by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.

Jan Dreier, Nikolas Mählmann, Sebastian Siebertz, and Szymon Toruńczyk. Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 125:1-125:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dreier_et_al:LIPIcs.ICALP.2023.125, author = {Dreier, Jan and M\"{a}hlmann, Nikolas and Siebertz, Sebastian and Toru\'{n}czyk, Szymon}, title = {{Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {125:1--125:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.125}, URN = {urn:nbn:de:0030-drops-181779}, doi = {10.4230/LIPIcs.ICALP.2023.125}, annote = {Keywords: stability, NIP, combinatorial characterization, first-order model checking} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results.
- For every monadically stable class C and fixed integer s ≥ 3, the Ramsey numbers R_C(s,t) are bounded from above by 𝒪(t^{s-1-δ}) for some δ > 0, improving the bound R(s,t) ∈ 𝒪(t^{s-1}/(log t)^{s-1}) known for the class of all graphs and the bounds known for k-stable graphs when s ≤ k.
- For every monadically stable class C and every integer r, there exists δ > 0 such that every graph G ∈ C that contains an r-subdivision of the biclique K_{t,t} as a subgraph also contains K_{t^δ,t^δ} as a subgraph. This generalizes earlier results for nowhere dense graph classes.
- We obtain a stronger regularity lemma for monadically stable classes of graphs.
- Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.

Jan Dreier, Nikolas Mählmann, Amer E. Mouawad, Sebastian Siebertz, and Alexandre Vigny. Combinatorial and Algorithmic Aspects of Monadic Stability. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dreier_et_al:LIPIcs.ISAAC.2022.11, author = {Dreier, Jan and M\"{a}hlmann, Nikolas and Mouawad, Amer E. and Siebertz, Sebastian and Vigny, Alexandre}, title = {{Combinatorial and Algorithmic Aspects of Monadic Stability}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.11}, URN = {urn:nbn:de:0030-drops-172961}, doi = {10.4230/LIPIcs.ISAAC.2022.11}, annote = {Keywords: Monadic Stability, Structural Graph Theory, Ramsey Numbers, Regularity, Kernels} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. Bounded backdoor size implies bounded backdoor depth, but there are formulas of constant backdoor depth and arbitrarily large backdoor size.
We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. We base our FPT approximation algorithm on a sophisticated notion of obstructions, extending Mählmann et al.’s obstruction trees in various ways, including the addition of separator obstructions. We develop the algorithm through a new game-theoretic framework that simplifies the reasoning about backdoors.
Finally, we show that bounded backdoor depth captures tractable classes of CNF formulas not captured by any known method.

Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. SAT Backdoors: Depth Beats Size. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 46:1-46:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dreier_et_al:LIPIcs.ESA.2022.46, author = {Dreier, Jan and Ordyniak, Sebastian and Szeider, Stefan}, title = {{SAT Backdoors: Depth Beats Size}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {46:1--46:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.46}, URN = {urn:nbn:de:0030-drops-169840}, doi = {10.4230/LIPIcs.ESA.2022.46}, annote = {Keywords: satisfiability, backdoor (depth)} }

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**Published in:** LIPIcs, Volume 235, 28th International Conference on Principles and Practice of Constraint Programming (CP 2022)

The constraint satisfaction problem (CSP) is among the most studied computational problems. While NP-hard, many tractable subproblems have been identified (Bulatov 2017, Zuk 2017). Backdoors, introduced by Williams, Gomes, and Selman (2003), gradually extend such a tractable class to all CSP instances of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a CSP instance and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021) for SAT, is a more refined distance measure, which admits the parallel utilization of different backdoor variables. Bounded backdoor size implies bounded backdoor depth, but there are instances of constant backdoor depth and arbitrarily large backdoor size. Dreier, Ordyniak, and Szeider (2022) provided fixed-parameter algorithms for finding backdoors of small depth into the classes of Horn and Krom formulas.
In this paper, we consider backdoor depth for CSP. We consider backdoors w.r.t. tractable subproblems C_Γ of the CSP defined by a constraint language Γ, i.e., where all the constraints use relations from the language Γ. Building upon Dreier et al.’s game-theoretic approach and their notion of separator obstructions, we show that for any finite, tractable, semi-conservative constraint language Γ, the CSP is fixed-parameter tractable parameterized by the backdoor depth into C_Γ plus the domain size.
With backdoors of low depth, we reach classes of instances that require backdoors of arbitrary large size. Hence, our results strictly generalize several known results for CSP that are based on backdoor size.

Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. CSP Beyond Tractable Constraint Languages. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 235, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dreier_et_al:LIPIcs.CP.2022.20, author = {Dreier, Jan and Ordyniak, Sebastian and Szeider, Stefan}, title = {{CSP Beyond Tractable Constraint Languages}}, booktitle = {28th International Conference on Principles and Practice of Constraint Programming (CP 2022)}, pages = {20:1--20:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-240-2}, ISSN = {1868-8969}, year = {2022}, volume = {235}, editor = {Solnon, Christine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2022.20}, URN = {urn:nbn:de:0030-drops-166490}, doi = {10.4230/LIPIcs.CP.2022.20}, annote = {Keywords: CSP, backdoor depth, constraint language, tractable class, recursive backdoor} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central role in today’s society. Algorithmic meta-theorems show that many problems can be solved efficiently. Since logic is a powerful tool to model problems, it has been used to obtain very general meta-theorems. In this work, we consider all problems definable in first-order logic and analyze which properties of complex networks allow them to be solved efficiently.
The mathematical tool to describe complex networks are random graph models. We define a property of random graph models called α-power-law-boundedness. Roughly speaking, a random graph is α-power-law-bounded if it does not admit strong clustering and its degree sequence is bounded by a power-law distribution with exponent at least α (i.e. the fraction of vertices with degree k is roughly O(k^{-α})).
We solve the first-order model-checking problem (parameterized by the length of the formula) in almost linear FPT time on random graph models satisfying this property with α ≥ 3. This means in particular that one can solve every problem expressible in first-order logic in almost linear expected time on these random graph models. This includes for example preferential attachment graphs, Chung-Lu graphs, configuration graphs, and sparse Erdős-Rényi graphs. Our results match known hardness results and generalize previous tractability results on this topic.

Jan Dreier, Philipp Kuinke, and Peter Rossmanith. First-Order Model-Checking in Random Graphs and Complex Networks. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 40:1-40:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dreier_et_al:LIPIcs.ESA.2020.40, author = {Dreier, Jan and Kuinke, Philipp and Rossmanith, Peter}, title = {{First-Order Model-Checking in Random Graphs and Complex Networks}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {40:1--40:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.40}, URN = {urn:nbn:de:0030-drops-129068}, doi = {10.4230/LIPIcs.ESA.2020.40}, annote = {Keywords: random graphs, average case analysis, first-order model-checking} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Preferential attachment graphs are random graphs designed to mimic properties of real word networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We prove various structural asymptotic properties of this graph model. In particular, we show that the size of the largest r-shallow clique minor in Gⁿ_m is at most log(n)^{O(r²)}m^{O(r)}. Furthermore, there exists a one-subdivided clique of size log(n)^{1/4}. Therefore, preferential attachment graphs are asymptotically almost surely somewhere dense and algorithmic techniques developed for structurally sparse graph classes are not directly applicable. However, they are just barely somewhere dense. The removal of just slightly more than a polylogarithmic number of vertices asymptotically almost surely yields a graph with locally bounded treewidth.

Jan Dreier, Philipp Kuinke, and Peter Rossmanith. Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dreier_et_al:LIPIcs.APPROX/RANDOM.2020.14, author = {Dreier, Jan and Kuinke, Philipp and Rossmanith, Peter}, title = {{Maximum Shallow Clique Minors in Preferential Attachment Graphs Have Polylogarithmic Size}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.14}, URN = {urn:nbn:de:0030-drops-126171}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.14}, annote = {Keywords: Random Graphs, Preferential Attachment, Sparsity, Somewhere Dense} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Many graph properties are expressible in first order logic. Whether a graph contains a clique or a dominating set of size k are two examples. For the solution size as its parameter the first one is W[1]-complete and the second one W[2]-complete meaning that both of them are hard problems in the worst-case. If we look at both problem from the aspect of average-case complexity, the picture changes. Clique can be solved in expected FPT time on uniformly distributed graphs of size n, while this is not clear for Dominating Set. We show that it is indeed unlikely that Dominating Set can be solved efficiently on random graphs: If yes, then every first-order expressible graph property can be solved in expected FPT time, too. Furthermore, this remains true when we consider random graphs with an arbitrary constant edge probability. We identify a very simple problem on random matrices that is equally hard to solve on average: Given a square boolean matrix, are there k rows whose logical AND is the zero vector? The related Even Set problem on the other hand turns out to be efficiently solvable on random instances, while it is known to be hard in the worst-case.

Jan Dreier, Henri Lotze, and Peter Rossmanith. Hard Problems on Random Graphs. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{dreier_et_al:LIPIcs.ICALP.2020.40, author = {Dreier, Jan and Lotze, Henri and Rossmanith, Peter}, title = {{Hard Problems on Random Graphs}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {40:1--40:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.40}, URN = {urn:nbn:de:0030-drops-124477}, doi = {10.4230/LIPIcs.ICALP.2020.40}, annote = {Keywords: random graphs, average-case complexity, first-order model checking} }

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**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem.
Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard.
Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs.

Jan Dreier, Janosch Fuchs, Tim A. Hartmann, Philipp Kuinke, Peter Rossmanith, Bjoern Tauer, and Hung-Lung Wang. The Complexity of Packing Edge-Disjoint Paths. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dreier_et_al:LIPIcs.IPEC.2019.10, author = {Dreier, Jan and Fuchs, Janosch and Hartmann, Tim A. and Kuinke, Philipp and Rossmanith, Peter and Tauer, Bjoern and Wang, Hung-Lung}, title = {{The Complexity of Packing Edge-Disjoint Paths}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {10:1--10:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.10}, URN = {urn:nbn:de:0030-drops-114710}, doi = {10.4230/LIPIcs.IPEC.2019.10}, annote = {Keywords: parameterized complexity, embedding, packing, covering, Hamiltonian path, unary binpacking, path-perfect graphs} }

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**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

It is known that FO model-checking is fixed-parameter tractable on Erdős - Rényi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon/n) for every epsilon>0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erdős - Rényi graphs with vertex colors the above stated upper bound by Grohe is the best possible.
More specifically, we show that there is no FO model-checking algorithm with average FPT run time on vertex-colored Erdős - Rényi graphs G(n,n^delta/n) (0 < delta < 1) unless AW[*]subseteq FPT/poly. This might be the first result where parameterized average-case intractability of a natural problem with a natural probability distribution is linked to worst-case complexity assumptions.
We further provide hardness results for FO model-checking on other random graph models, including G(n,1/2) and Chung-Lu graphs, where our intractability results tightly match known tractability results [E. D. Demaine et al., 2014]. We also provide lower bounds on the size of shallow clique minors in certain Erdős - Rényi and Chung - Lu graphs.

Jan Dreier and Peter Rossmanith. Hardness of FO Model-Checking on Random Graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dreier_et_al:LIPIcs.IPEC.2019.11, author = {Dreier, Jan and Rossmanith, Peter}, title = {{Hardness of FO Model-Checking on Random Graphs}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {11:1--11:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.11}, URN = {urn:nbn:de:0030-drops-114721}, doi = {10.4230/LIPIcs.IPEC.2019.11}, annote = {Keywords: random graphs, FO model-checking} }

Document

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Network motifs are small patterns that occur in a network significantly more often than expected. They have gathered a lot of interest, as they may describe functional dependencies of complex networks and yield insights into their basic structure [Milo et al., 2002]. Therefore, a large amount of work went into the development of methods for network motif detection in complex networks [Kashtan et al., 2004; Schreiber and Schwöbbermeyer, 2005; Chen et al., 2006; Wernicke, 2006; Grochow and Kellis, 2007; Alon et al., 2008; Omidi et al., 2009]. The underlying problem of motif detection is to count how often a copy of a pattern graph H occurs in a target graph G. This problem is #W[1]-hard when parameterized by the size of H [Flum and Grohe, 2004] and cannot be solved in time f(|H|)n^o(|H|) under #ETH [Chen et al., 2005].
Preferential attachment graphs [Barabási and Albert, 1999] are a very popular random graph model designed to mimic complex networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. Preferential attachment has been empirically observed in real growing networks [Newman, 2001; Jeong et al., 2003].
We show that one can count subgraph copies of a graph H in the preferential attachment graph G^n_m (with n vertices and nm edges, where m is usually a small constant) in expected time f(|H|) m^O(|H|^6) log(n)^O(|H|^12) n. This means the motif counting problem can be solved in expected quasilinear FPT time on preferential attachment graphs with respect to the parameters |H| and m. In particular, for fixed H and m the expected run time is O(n^(1+epsilon)) for every epsilon>0.
Our results are obtained using new concentration bounds for degrees in preferential attachment graphs. Assume the (total) degree of a set of vertices at a time t of the random process is d. We show that if d is sufficiently large then the degree of the same set at a later time n is likely to be in the interval (1 +/- epsilon)d sqrt(n/t) (for epsilon > 0) for all n >= t. More specifically, the probability that this interval is left is exponentially small in d.

Jan Dreier and Peter Rossmanith. Motif Counting in Preferential Attachment Graphs. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dreier_et_al:LIPIcs.FSTTCS.2019.13, author = {Dreier, Jan and Rossmanith, Peter}, title = {{Motif Counting in Preferential Attachment Graphs}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.13}, URN = {urn:nbn:de:0030-drops-115750}, doi = {10.4230/LIPIcs.FSTTCS.2019.13}, annote = {Keywords: random graphs, motif counting, average case analysis, preferential attachment graphs} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

In Conspiracy Santa, a variant of Secret Santa, a group of people offer each other Christmas gifts, where each member of the group receives a gift from the other members of the group. To that end, the members of the group form conspiracies, to decide on appropriate gifts, and usually divide the cost of each gift among all participants of that conspiracy. This requires to settle the shared expenses per conspiracy, so Conspiracy Santa can actually be seen as an aggregation of several shared expenses problems.
First, we show that the problem of finding a minimal number of transaction when settling shared expenses is NP-complete. Still, there exists good greedy approximations. Second, we present a greedy distributed secure solution to Conspiracy Santa. This solution allows a group of people to share the expenses for the gifts in such a way that no participant learns the price of his gift, but at the same time notably reduces the number of transactions with respect to a naive aggregation. Furthermore, our solution does not require a trusted third party, and can either be implemented physically (the participants are in the same room and exchange money using envelopes) or, virtually, using a cryptocurrency.

Xavier Bultel, Jannik Dreier, Jean-Guillaume Dumas, and Pascal Lafourcade. A Cryptographer's Conspiracy Santa. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bultel_et_al:LIPIcs.FUN.2018.13, author = {Bultel, Xavier and Dreier, Jannik and Dumas, Jean-Guillaume and Lafourcade, Pascal}, title = {{A Cryptographer's Conspiracy Santa}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.13}, URN = {urn:nbn:de:0030-drops-88047}, doi = {10.4230/LIPIcs.FUN.2018.13}, annote = {Keywords: Secret Santa, Conspiracy Santa, Secure Multi-Party Computation, Cryptocurrency, Physical Cryptography} }

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**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Akari, Takuzu, Kakuro and KenKen are logic games similar to Sudoku. In Akari, a labyrinth on a grid has to be lit by placing lanterns, respecting various constraints. In Takuzu a grid has to be filled with 0's and 1's, while respecting certain constraints. In Kakuro a grid has to be filled with numbers such that the sums per row and column match given values; similarly in KenKen a grid has to be filled with numbers such that in given areas the product, sum, difference or quotient equals a given value. We give physical algorithms to realize zero-knowledge proofs for these games which allow a player to show that he knows a solution without revealing it. These interactive proofs can be realized with simple office material as they only rely on cards and envelopes. Moreover, we formalize our algorithms and prove their security.

Xavier Bultel, Jannik Dreier, Jean-Guillaume Dumas, and Pascal Lafourcade. Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bultel_et_al:LIPIcs.FUN.2016.8, author = {Bultel, Xavier and Dreier, Jannik and Dumas, Jean-Guillaume and Lafourcade, Pascal}, title = {{Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {8:1--8:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.8}, URN = {urn:nbn:de:0030-drops-58693}, doi = {10.4230/LIPIcs.FUN.2016.8}, annote = {Keywords: Physical Cryptography, ZKP, Games, Akari, Kakuro, KenKen, Takuzu} }