Document

**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

For some time, it has been known that the model checking problem for first-order formulas is fixed-parameter tractable on nowhere dense graph classes, so we shall ask in which direction there is space for improvements. One of the possible directions is to go beyond first-order formulas: Augmenting first-order logic with general counting quantifiers increases the expressiveness by far, but makes the model checking problem hard even on graphs of bounded tree-depth. The picture is different if we allow only "simple" - but arbitrarily nested - counting terms of the form #y φ(x^- ,y) > N. Even then, only approximate model checking is possible on graph classes of bounded expansion. Here, the largest known logic fragment, on which exact model checking is still fpt, consists of formulas of the form ∃x_1 … ∃x_k #y φ(x^- ,y) > N, where φ(x^- ,y) is a first-order formula without counting terms. An example of a problem that can be expressed in this way is partial dominating set: Are there k vertices that dominate at least a given number of vertices in the graph? The complexity of the same problem is open if you replace at least with exactly. Likewise, the complexity of "are there k vertices that dominate at least half of the blue and half of the red vertices?" is also open. We answer both questions by providing an fpt algorithm that solves the model checking problem for formulas of the more general form ψ ≡ ∃x_1 … ∃x_k P(#y φ_1(x^- ,y), …, #y φ_ℓ(x^- ,y)), where P is an arbitrary polynomially computable predicate on numbers. The running time is f(|ψ|)n^{𝓁+1} polylog(n) on graph classes of bounded expansion. Under SETH, this running time is tight up to almost linear factor.

Daniel Mock and Peter Rossmanith. Solving a Family Of Multivariate Optimization and Decision Problems on Classes of Bounded Expansion. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 35:1-35:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mock_et_al:LIPIcs.SWAT.2024.35, author = {Mock, Daniel and Rossmanith, Peter}, title = {{Solving a Family Of Multivariate Optimization and Decision Problems on Classes of Bounded Expansion}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {35:1--35:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.35}, URN = {urn:nbn:de:0030-drops-200754}, doi = {10.4230/LIPIcs.SWAT.2024.35}, annote = {Keywords: bounded expansion, parameterized algorithms, sparsity, counting logic, dominating set, model checking, multivariate optimization} }

Document

**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

In the proportional knapsack problem, we are given a knapsack of some capacity and a set of variably sized items. The goal is to pack a selection of these items that fills the knapsack as much as possible. The online version of this problem reveals the items and their sizes not all at once but one by one. For each item, the algorithm has to decide immediately whether to pack it or not. We consider a natural variant of this online knapsack problem, which has been coined removable knapsack. It differs from the classical variant by allowing the removal of any packed item from the knapsack. Repacking is impossible, however: Once an item is removed, it is gone for good.
We analyze the advice complexity of this problem. It measures how many advice bits an omniscient oracle needs to provide for an online algorithm to reach any given competitive ratio, which is - understood in its strict sense - just the algorithm’s approximation factor. The online knapsack problem is known for its peculiar advice behavior involving three jumps in competitivity. We show that the advice complexity of the version with removability is quite different but just as interesting: The competitivity starts from the golden ratio when no advice is given. It then drops down to 1+ε for a constant amount of advice already, which requires logarithmic advice in the classical version. Removability comes as no relief to the perfectionist, however: Optimality still requires linear advice as before. These results are particularly noteworthy from a structural viewpoint for the exceptionally slow transition from near-optimality to optimality.
Our most important and demanding result shows that the general knapsack problem, which allows an item’s value to differ from its size, exhibits a similar behavior for removability, but with an even more pronounced jump from an unbounded competitive ratio to near-optimality within just constantly many advice bits. This is a unique behavior among the problems considered in the literature so far.
An advice analysis is interesting in its own right, as it allows us to measure the information content of a problem and leads to structural insights. But it also provides insurmountable lower bounds, applicable to any kind of additional information about the instances, including predictions provided by machine-learning algorithms and artificial intelligence. Unexpectedly, advice algorithms are useful in various real-life situations, too. For example, they provide smart strategies for cooperation in winner-take-all competitions, where several participants pool together to implement different strategies and share the obtained prize. Further illustrating the versatility of our advice-complexity bounds, our results automatically improve some of the best known lower bounds on the competitive ratio for removable knapsack with randomization. The presented advice algorithms also automatically yield deterministic algorithms for established deterministic models such as knapsack with a resource buffer and various problems with more than one knapsack. In their seminal paper introducing removability to the knapsack problem, Iwama and Taketomi have indeed proposed a multiple knapsack problem for which we can establish a one-to-one correspondence with the advice model; this paper therefore even provides a comprehensive analysis for this up until now neglected problem.

Hans-Joachim Böckenhauer, Fabian Frei, and Peter Rossmanith. Removable Online Knapsack and Advice. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bockenhauer_et_al:LIPIcs.STACS.2024.18, author = {B\"{o}ckenhauer, Hans-Joachim and Frei, Fabian and Rossmanith, Peter}, title = {{Removable Online Knapsack and Advice}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {18:1--18:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.18}, URN = {urn:nbn:de:0030-drops-197283}, doi = {10.4230/LIPIcs.STACS.2024.18}, annote = {Keywords: Removable Online Knapsack, Multiple Knapsack, Advice Analysis, Advice Applications, Machine Learning and AI} }

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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

In the Online Simple Knapsack problem, an algorithm has to pack a knapsack of unit size as full as possible with items that arrive sequentially. The algorithm has no prior knowledge of the length or nature of the instance. Its performance is then measured against the best possible packing of all items of the same instance, over all possible instances.
In the classical model for online computation, it is well known that there exists no constant bound for the ratio between the size of an optimal packing and the size of an online algorithm’s packing. A recent variation of the classical online model is that of predictions. In this model, an algorithm is given knowledge about the instance in advance, which is in reality distorted by some factor δ that is commonly unknown to the algorithm. The algorithm only learns about the actual nature of the elements of an input once they are revealed and an irrevocable and immediate decision has to be made. In this work, we study a slight variation of this model in which the error term, and thus the range of sizes that an announced item may actually lay in, is given to the algorithm in advance. It thus knows the range of sizes from which the actual size of each item is selected from.
We find that the analysis of the Online Simple Knapsack problem under this model is surprisingly involved. For values of 0 < δ ≤ 1/7, we prove a tight competitive ratio of 2. From there on, we are able to prove that there are at least three alternating functions that describe the competitive ratio. We provide partially tight bounds for the whole range of 0 < δ < 1, showing in particular that the function of the competitive ratio depending on δ is not continuous.

Matthias Gehnen, Henri Lotze, and Peter Rossmanith. Online Simple Knapsack with Bounded Predictions. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gehnen_et_al:LIPIcs.STACS.2024.37, author = {Gehnen, Matthias and Lotze, Henri and Rossmanith, Peter}, title = {{Online Simple Knapsack with Bounded Predictions}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {37:1--37:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.37}, URN = {urn:nbn:de:0030-drops-197476}, doi = {10.4230/LIPIcs.STACS.2024.37}, annote = {Keywords: Online problem, Simple Knapsack, Predictions, Machine-Learned Advice} }

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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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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

In the online simple knapsack problem, a knapsack of unit size 1 is given and an algorithm is tasked to fill it using a set of items that are revealed one after another. Each item must be accepted or rejected at the time they are presented, and these decisions are irrevocable. No prior knowledge about the set and sequence of items is given. The goal is then to maximize the sum of the sizes of all packed items compared to an optimal packing of all items of the sequence.
In this paper, we combine two existing variants of the problem that each extend the range of possible actions for a newly presented item by a new option. The first is removability, in which an item that was previously packed into the knapsack may be finally discarded at any point. The second is reservations, which allows the algorithm to delay the decision on accepting or rejecting a new item indefinitely for a proportional fee relative to the size of the given item.
If both removability and reservations are permitted, we show that the competitive ratio of the online simple knapsack problem rises depending on the relative reservation costs. As soon as any nonzero fee has to be paid for a reservation, no online algorithm can be better than 1.5-competitive. With rising reservation costs, this competitive ratio increases up to the golden ratio (ϕ ≈ 1.618) that is reached for relative reservation costs of 1-√5/3 ≈ 0.254. We provide a matching upper and lower bound for relative reservation costs up to this value. From this point onward, the tight bound by Iwama and Taketomi for the removable knapsack problem is the best possible competitive ratio, not using any reservations.

Elisabet Burjons, Matthias Gehnen, Henri Lotze, Daniel Mock, and Peter Rossmanith. The Online Simple Knapsack Problem with Reservation and Removability. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 29:1-29:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{burjons_et_al:LIPIcs.MFCS.2023.29, author = {Burjons, Elisabet and Gehnen, Matthias and Lotze, Henri and Mock, Daniel and Rossmanith, Peter}, title = {{The Online Simple Knapsack Problem with Reservation and Removability}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {29:1--29:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.29}, URN = {urn:nbn:de:0030-drops-185635}, doi = {10.4230/LIPIcs.MFCS.2023.29}, annote = {Keywords: online algorithm, knapsack, competitive ratio, reservation, preemption} }

Document

**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

The non-existence of polynomial kernels for OR- and AND-compositional problems is now a well-established result. Some of these problems have adaptive or non-adaptive polynomial Turing kernels. Up to now, most known polynomial Turing kernels are non-adaptive and most of them are of the conjunctive or disjunctive kind. For some problems it has been conjectured that the existence of polynomial Turing kernels is unlikely. For instance, either all or none of the WK[1]-complete problems have polynomial Turing kernels. While it has been conjectured that they do not, a proof tying their non-existence to some complexity theoretic assumption is still missing and seems to be beyond the reach of today’s standard techniques.
In this paper, we prove that OR-compositional problems and all WK[1]-hard problems do not have conjunctive polynomial kernels, a special type of non-adaptive Turing kernels, under the assumption that coNP ⊈ NP/poly. Similarly, it is unlikely that AND-compositional problems have disjunctive polynomial kernels. Moreover, we present a way to prove that the parameterized versions of some ⊕ P-hard problems, for instance, Odd Path on planar graphs, do not have conjunctive or disjunctive polynomial kernels, unless coNP ⊆ NP/poly. Finally, we identify a problem that is unlikely to have either a conjunctive or disjunctive polynomial kernel, unless coNP ⊆ NP/poly, due to a reduction from an NP-hard problem instead: Weighted Odd Path on planar graphs.

Elisabet Burjons and Peter Rossmanith. Lower Bounds for Conjunctive and Disjunctive Turing Kernels. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{burjons_et_al:LIPIcs.IPEC.2021.12, author = {Burjons, Elisabet and Rossmanith, Peter}, title = {{Lower Bounds for Conjunctive and Disjunctive Turing Kernels}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.12}, URN = {urn:nbn:de:0030-drops-153953}, doi = {10.4230/LIPIcs.IPEC.2021.12}, annote = {Keywords: Parameterized Complexity, Turing kernels} }

Document

**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

In the Online Simple Knapsack Problem we are given a knapsack of unit size 1. Items of size smaller or equal to 1 are presented in an iterative fashion and an algorithm has to decide whether to permanently reject or include each item into the knapsack without any knowledge about the rest of the instance. The goal is then to pack the knapsack as full as possible. In this work, we introduce a third option additional to those of packing and rejecting an item, namely that of reserving an item for the cost of a fixed fraction α of its size. An algorithm may pay this fraction in order to postpone its decision on whether to include or reject the item until after the last item of the instance was presented.
While the classical Online Simple Knapsack Problem does not admit any constantly bounded competitive ratio in the deterministic setting, we find that adding the possibility of reservation makes the problem constantly competitive, with varying competitive ratios depending on the value of α. We give upper and lower bounds for the whole range of reservation costs, with tight bounds for costs up to 1/6 - an area that is strictly 2-competitive - , for costs between √2-1 and 1 - an area that is strictly (2+α)-competitive up to ϕ -1, and strictly 1/(1-α)-competitive above ϕ-1, where ϕ is the golden ratio.
With our analysis, we find a counterintuitive characteristic of the problem: Intuitively, one would expect that the possibility of rejecting items becomes more and more helpful for an online algorithm with growing reservation costs. However, for higher reservation costs above √2-1, an algorithm that is unable to reject any items tightly matches the lower bound and is thus the best possible. On the other hand, for any positive reservation cost smaller than 1/6, any algorithm that is unable to reject any items performs considerably worse than one that is able to reject.

Hans-Joachim Böckenhauer, Elisabet Burjons, Juraj Hromkovič, Henri Lotze, and Peter Rossmanith. Online Simple Knapsack with Reservation Costs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bockenhauer_et_al:LIPIcs.STACS.2021.16, author = {B\"{o}ckenhauer, Hans-Joachim and Burjons, Elisabet and Hromkovi\v{c}, Juraj and Lotze, Henri and Rossmanith, Peter}, title = {{Online Simple Knapsack with Reservation Costs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {16:1--16:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.16}, URN = {urn:nbn:de:0030-drops-136613}, doi = {10.4230/LIPIcs.STACS.2021.16}, annote = {Keywords: Online problem, Simple knapsack, Reservation costs} }

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**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

We analyze a little riddle that has challenged mathematicians for half a century.
Imagine three clubs catering to people with some niche interest. Everyone willing to join a club has done so and nobody new will pick up this eccentric hobby for the foreseeable future, thus the mutually exclusive clubs compete for a common constituency. Members are highly invested in their chosen club; only a targeted campaign plus prolonged personal persuasion can convince them to consider switching. Even then, they will never be enticed into a bigger group as they naturally pride themselves in avoiding the mainstream. Therefore each club occasionally starts a campaign against a larger competitor and sends its own members out on a recommendation program. Each will win one person over; the small club can thus effectively double its own numbers at the larger one’s expense.
Is there always a risk for one club to wind up with zero members, forcing it out of business? If so, how many campaign cycles will this take?

Fabian Frei, Peter Rossmanith, and David Wehner. An Open Pouring Problem. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 14:1-14:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{frei_et_al:LIPIcs.FUN.2021.14, author = {Frei, Fabian and Rossmanith, Peter and Wehner, David}, title = {{An Open Pouring Problem}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {14:1--14:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.14}, URN = {urn:nbn:de:0030-drops-127751}, doi = {10.4230/LIPIcs.FUN.2021.14}, annote = {Keywords: Pitcher Pouring Problem, Water Jug Riddle, Water Bucket Problem, Vessel Puzzle, Complexity, Die Hard} }

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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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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization.

Pavol Ďuriš, Rastislav Královič, Richard Královič, Dana Pardubská, Martin Pašen, and Peter Rossmanith. Randomization in Non-Uniform Finite Automata. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 30:1-30:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{duris_et_al:LIPIcs.MFCS.2020.30, author = {\v{D}uri\v{s}, Pavol and Kr\'{a}lovi\v{c}, Rastislav and Kr\'{a}lovi\v{c}, Richard and Pardubsk\'{a}, Dana and Pa\v{s}en, Martin and Rossmanith, Peter}, title = {{Randomization in Non-Uniform Finite Automata}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {30:1--30:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.30}, URN = {urn:nbn:de:0030-drops-126987}, doi = {10.4230/LIPIcs.MFCS.2020.30}, annote = {Keywords: finite automata, non-uniform computation, randomization} }

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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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Complete Volume

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

LIPIcs, Volume 138, MFCS'19, Complete Volume

44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{rossmanith_et_al:LIPIcs.MFCS.2019, title = {{LIPIcs, Volume 138, MFCS'19, Complete Volume}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019}, URN = {urn:nbn:de:0030-drops-112092}, doi = {10.4230/LIPIcs.MFCS.2019}, annote = {Keywords: Theory of computation} }

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Front Matter

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Front Matter, Table of Contents, Preface, Conference Organization

44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{rossmanith_et_al:LIPIcs.MFCS.2019.0, author = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.0}, URN = {urn:nbn:de:0030-drops-109444}, doi = {10.4230/LIPIcs.MFCS.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Given a graph G=(V, E), an s-plex S\subseteq V is a vertex subset such that for v\in S the degree of v in G[S] is at least |S|-s. An s-plex bipartition \mathcal{P}=(V_1, V_2) is a bipartition of G=(V, E), V=V_1\uplus V_2, satisfying that both V_1 and V_2 are s-plexes. Given an instance G=(V, E) and a parameter k, the s-Plex Bipartition problem asks whether there exists an s-plex bipartition of G such that min{|V_1|, |V_2|\}\leq k. The s-Plex Bipartition problem is NP-complete. However, it is still open whether this problem is fixed-parameter tractable. In this paper, we give a fixed-parameter algorithm for 2-Plex Bipartition running in time O*(2.4143^k). A graph G = (V, E) is called defective (p, d)-colorable if it admits a vertex coloring with p colors such that each color class in G induces a subgraph of maximum degree at most d. A graph G admits an s-plex bipartition if and only if the complement graph of G, \bar{G}, admits a defective (2, s-1)-coloring such that one of the two color classes is of size at most k. By applying our fixed-parameter algorithm as a subroutine, one can find a defective (2,1)-coloring with one of the two colors of minimum cardinality for a given graph in O*(1.5539^n) time where n is the number of vertices in the input graph.

Li-Hsuan Chen, Sun-Yuan Hsieh, Ling-Ju Hung, and Peter Rossmanith. An Efficient Fixed-Parameter Algorithm for the 2-Plex Bipartition Problem. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 20:1-20:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chen_et_al:LIPIcs.ISAAC.2017.20, author = {Chen, Li-Hsuan and Hsieh, Sun-Yuan and Hung, Ling-Ju and Rossmanith, Peter}, title = {{An Efficient Fixed-Parameter Algorithm for the 2-Plex Bipartition Problem}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {20:1--20:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.20}, URN = {urn:nbn:de:0030-drops-82458}, doi = {10.4230/LIPIcs.ISAAC.2017.20}, annote = {Keywords: 2-plex, 2-plex bipartition, bounded-degree-1 set bipartition, defective (2,1)-coloring} }

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**Published in:** Dagstuhl Reports, Volume 4, Issue 11 (2015)

This report documents the program and the outcomes of Dagstuhl Seminar 14451 "Optimality and tight results in parameterized complexity". Over the last two decades parameterized complexity has become one of the main tools for handling intractable problems. Recently, tools have been developed not only to classify problems, but also to make statements about how close an algorithm is to being optimal with respect to running time. The focus of this seminar is to highlight and discuss recent, relevant
results within this optimality framework and discover fruitful research directions. The report contains the abstracts of the results presented at the seminar, as well as a collection of open problems stated at the seminar.

Stefan Kratsch, Daniel Lokshtanov, Dániel Marx, and Peter Rossmanith. Optimality and tight results in parameterized complexity (Dagstuhl Seminar 14451). In Dagstuhl Reports, Volume 4, Issue 11, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{kratsch_et_al:DagRep.4.11.1, author = {Kratsch, Stefan and Lokshtanov, Daniel and Marx, D\'{a}niel and Rossmanith, Peter}, title = {{Optimality and tight results in parameterized complexity (Dagstuhl Seminar 14451)}}, pages = {1--21}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2015}, volume = {4}, number = {11}, editor = {Kratsch, Stefan and Lokshtanov, Daniel and Marx, D\'{a}niel and Rossmanith, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.11.1}, URN = {urn:nbn:de:0030-drops-49677}, doi = {10.4230/DagRep.4.11.1}, annote = {Keywords: Algorithms, parameterized complexity, kernels, width measures, exponential time hypothesis, lower bounds} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

One of the most important algorithmic meta-theorems is a famous result
by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that
even MSO1 model-checking with a fixed set of vertex labels,
but without edge-set quantifications, is not in XP wrt the formula
size as parameter for graph classes which are subgraph-closed and
whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no
subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326, author = {Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath}, title = {{Lower Bounds on the Complexity of MSO\underline1 Model-Checking}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {326--337}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326}, URN = {urn:nbn:de:0030-drops-34185}, doi = {10.4230/LIPIcs.STACS.2012.326}, annote = {Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We present a simple exact algorithm for the \is\ problem with a runtime bounded
by $O(\rt^n \poly(n))$. This bound is obtained by, firstly, applying a new
branching rule and, secondly, by a distinct, computer-aided case analysis.
The new branching rule uses the concept of satellites and has previously only
been used in an algorithm for sparse graphs.
The computer-aided case analysis allows us to capture the behavior of our algorithm
in more detail than in a traditional analysis.
The main purpose of this paper is to demonstrate how a very simple
algorithm can outperform more complicated ones if the right analysis
of its running time is performed.

Joachim Kneis, Alexander Langer, and Peter Rossmanith. A Fine-grained Analysis of a Simple Independent Set Algorithm. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 287-298, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{kneis_et_al:LIPIcs.FSTTCS.2009.2326, author = {Kneis, Joachim and Langer, Alexander and Rossmanith, Peter}, title = {{A Fine-grained Analysis of a Simple Independent Set Algorithm}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {287--298}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2326}, URN = {urn:nbn:de:0030-drops-23269}, doi = {10.4230/LIPIcs.FSTTCS.2009.2326}, annote = {Keywords: Exact Algorithms, Independent Set, Computer-aided Proof} }

Document

**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

Rodney G. Downey, Michael R. Fellows, Rolf Niedermeier, and Peter Rossmanith. Parameterized Complexity (Dagstuhl Seminar 01311). Dagstuhl Seminar Report 316, pp. 1-28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2002)

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@TechReport{downey_et_al:DagSemRep.316, author = {Downey, Rodney G. and Fellows, Michael R. and Niedermeier, Rolf and Rossmanith, Peter}, title = {{Parameterized Complexity (Dagstuhl Seminar 01311)}}, pages = {1--28}, ISSN = {1619-0203}, year = {2002}, type = {Dagstuhl Seminar Report}, number = {316}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.316}, URN = {urn:nbn:de:0030-drops-152009}, doi = {10.4230/DagSemRep.316}, }

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