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

The concept of Roman domination has recently been studied concerning enumerating and counting in F. N. Abu-Khzam et al. (WG 2022). More technically speaking, a function that assigns 0,1,2 to the vertices of an undirected graph is called a Roman dominating function if each vertex assigned zero has a neighbor assigned two. Such a function is called minimal if decreasing any assignment to any vertex would yield a function that is no longer a Roman dominating function. It has been shown that minimal Roman dominating functions can be enumerated with polynomial delay, i.e., between any two outputs of a solution, no more than polynomial time will elapse. This contrasts what is known about minimal dominating sets, where the question whether or not these can be enumerated with polynomial delay is open for more than 40 years. This makes the concept of Roman domination rather special and interesting among the many variants of domination problems studied in the literature, as it has been shown for several of these variants that the question of enumerating minimal solutions is tightly linked to that of enumerating minimal dominating sets, see M. Kanté et al. in SIAM J. Disc. Math., 2014. The running time of the mentioned enumeration algorithm for minimal Roman dominating functions (Abu-Khzam et al., WG 2022) could be estimated as 𝒪(1.9332ⁿ) on general graphs of order n. Here, we focus on special graph classes, as has been also done for enumerating minimal dominating sets before. More specifically, for chordal graphs, we present an enumeration algorithm running in time 𝒪(1.8940ⁿ). It is unknown if this gives a tight bound on the maximum number of minimal Roman dominating functions in chordal graphs. For interval graphs, we can lower this time bound further to 𝒪(1.7321ⁿ), which also matches the known lower bound concerning the maximum number of minimal Roman dominating functions. We can also provide a matching lower and upper bound for forests, which is (incidentally) the same, namely 𝒪^*(√3ⁿ). Furthermore, we present an optimal enumeration algorithm running in time 𝒪^*(∛3ⁿ) for split graphs and for cobipartite graphs, i.e., we can also give a matching lower bound example for these graph classes. Hence, our enumeration algorithms for interval graphs, forests, split graphs and cobipartite graphs are all optimal. The importance of our results stems from the fact that, for other types of domination problems, optimal enumeration algorithms are not always found.
Interestingly, we use a different form of analysis for the running times of our different algorithms, and the branchings had to be tailored and tweaked to obtain the intended optimality results. Our Roman dominating functions enumeration algorithm for trees and forests is distinctively different from the one for minimal dominating sets by Rote (SODA 2019).Our approach also allows to give concrete formulas for counting minimal Roman dominating functions on more concrete graph families like paths.

Faisal N. Abu-Khzam, Henning Fernau, and Kevin Mann. Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abukhzam_et_al:LIPIcs.MFCS.2023.6, author = {Abu-Khzam, Faisal N. and Fernau, Henning and Mann, Kevin}, title = {{Roman Census: Enumerating and Counting Roman Dominating Functions on Graph Classes}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {6:1--6:15}, 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.2023.6}, URN = {urn:nbn:de:0030-drops-185400}, doi = {10.4230/LIPIcs.MFCS.2023.6}, annote = {Keywords: special graph classes, counting problems, enumeration problems, domination problems, Roman domination} }

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

The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2ⁿ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time 𝒪(1.9896ⁿ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time 𝒪(1.9767ⁿ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890ⁿ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422ⁿ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195ⁿ) and Ω(1.4723ⁿ), respectively.
We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U|. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.

Faisal N. Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, and Kevin Mann. Enumerating Minimal Connected Dominating Sets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{abukhzam_et_al:LIPIcs.ESA.2022.1, author = {Abu-Khzam, Faisal N. and Fernau, Henning and Gras, Benjamin and Liedloff, Mathieu and Mann, Kevin}, title = {{Enumerating Minimal Connected Dominating Sets}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {1:1--1:15}, 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.2022.1}, URN = {urn:nbn:de:0030-drops-169390}, doi = {10.4230/LIPIcs.ESA.2022.1}, annote = {Keywords: enumeration problems, connected domination, degenerate graphs} }

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**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

We consider the unlabeled motion-planning problem of m unit-disc robots moving in a simple polygonal workspace of n edges. The goal is to find a motion plan that moves the robots to a given set of m target positions. For the unlabeled variant, it does not matter which robot reaches which target position as long as all target positions are occupied in the end.
If the workspace has narrow passages such that the robots cannot fit through them, then the free configuration space, representing all possible unobstructed positions of the robots, will consist of multiple connected components. Even if in each component of the free space the number of targets matches the number of start positions, the motion-planning problem does not always have a solution when the robots and their targets are positioned very densely. In this paper, we prove tight bounds on how much separation between start and target positions is necessary to always guarantee a solution. Moreover, we describe an algorithm that always finds a solution in time O(n log n + mn + m²) if the separation bounds are met. Specifically, we prove that the following separation is sufficient: any two start positions are at least distance 4 apart, any two target positions are at least distance 4 apart, and any pair of a start and a target positions is at least distance 3 apart. We further show that when the free space consists of a single connected component, the separation between start and target positions is not necessary.

Bahareh Banyassady, Mark de Berg, Karl Bringmann, Kevin Buchin, Henning Fernau, Dan Halperin, Irina Kostitsyna, Yoshio Okamoto, and Stijn Slot. Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{banyassady_et_al:LIPIcs.SoCG.2022.12, author = {Banyassady, Bahareh and de Berg, Mark and Bringmann, Karl and Buchin, Kevin and Fernau, Henning and Halperin, Dan and Kostitsyna, Irina and Okamoto, Yoshio and Slot, Stijn}, title = {{Unlabeled Multi-Robot Motion Planning with Tighter Separation Bounds}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.12}, URN = {urn:nbn:de:0030-drops-160203}, doi = {10.4230/LIPIcs.SoCG.2022.12}, annote = {Keywords: motion planning, computational geometry, simple polygon} }

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**Published in:** LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

The notion of synchronization of finite automata is connected to one of the long-standing open problems in combinatorial automata theory, which is Černý’s Conjecture. In this paper, we focus on so-called synchronization games. We will discuss how to present synchronization questions in a playful way. This leads us to study related complexity questions on certain classes of finite automata. More precisely, we consider weakly acyclic, commutative and k-simple idempotent automata. We encounter a number of complexity classes, ranging from L up to PSPACE.

Henning Fernau, Carolina Haase, and Stefan Hoffmann. The Synchronization Game on Subclasses of Automata. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fernau_et_al:LIPIcs.FUN.2022.14, author = {Fernau, Henning and Haase, Carolina and Hoffmann, Stefan}, title = {{The Synchronization Game on Subclasses of Automata}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {14:1--14:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-232-7}, ISSN = {1868-8969}, year = {2022}, volume = {226}, editor = {Fraigniaud, Pierre and Uno, Yushi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.14}, URN = {urn:nbn:de:0030-drops-159842}, doi = {10.4230/LIPIcs.FUN.2022.14}, annote = {Keywords: Synchronization of finite automata, computational complexity} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

In the Intersection Non-emptiness problem, we are given a list of finite automata A_1, A_2,… , A_m over a common alphabet Σ as input, and the goal is to determine whether some string w ∈ Σ^* lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels B_0 or B_{1/2} and already PSPACE-hard when all automata accept a language from the level B_1. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC^0 when restricted to level L_0; complete for L or NL, depending on the input representation, when restricted to languages in the level L_{1/2}; NP-complete when the input is given as DFAs accepting a language in L_1 or L_{3/2}; and finally, PSPACE-complete when the input automata accept languages in level L_2 or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in L_1 or L_{3/2} does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.

Emmanuel Arrighi, Henning Fernau, Stefan Hoffmann, Markus Holzer, Ismaël Jecker, Mateus de Oliveira Oliveira, and Petra Wolf. On the Complexity of Intersection Non-emptiness for Star-Free Language Classes. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.FSTTCS.2021.34, author = {Arrighi, Emmanuel and Fernau, Henning and Hoffmann, Stefan and Holzer, Markus and Jecker, Isma\"{e}l and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{On the Complexity of Intersection Non-emptiness for Star-Free Language Classes}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.34}, URN = {urn:nbn:de:0030-drops-155456}, doi = {10.4230/LIPIcs.FSTTCS.2021.34}, annote = {Keywords: Intersection Non-emptiness Problem, Star-Free Languages, Straubing-Th\'{e}rien Hierarchy, dot-depth Hierarchy, Commutative Languages, Complexity} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In this work, we consider the following order reconfiguration problem: Given a graph G together with linear orders ω and ω' of the vertices of G, can one transform ω into ω' by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most k? We show that this problem always has an affirmative answer when the input linear orders ω and ω' have cutwidth (pathwidth) at most k/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.

Emmanuel Arrighi, Henning Fernau, Mateus de Oliveira Oliveira, and Petra Wolf. Order Reconfiguration Under Width Constraints. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.MFCS.2021.8, author = {Arrighi, Emmanuel and Fernau, Henning and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{Order Reconfiguration Under Width Constraints}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.8}, URN = {urn:nbn:de:0030-drops-144486}, doi = {10.4230/LIPIcs.MFCS.2021.8}, annote = {Keywords: Parameterized Complexity, Order Reconfiguration, String Rewriting Systems} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We are studying a weighted version of a linear extension problem, given some finite partial order ρ, called Completion of an Ordering. While this problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, or computer memory management. Each application yields a special ρ. We also relate the interval width of ρ to parameterizations such as maximum range that have been introduced earlier in these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems.

Emmanuel Arrighi, Henning Fernau, Mateus de Oliveira Oliveira, and Petra Wolf. Width Notions for Ordering-Related Problems. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arrighi_et_al:LIPIcs.FSTTCS.2020.9, author = {Arrighi, Emmanuel and Fernau, Henning and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{Width Notions for Ordering-Related Problems}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {9:1--9:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.9}, URN = {urn:nbn:de:0030-drops-132505}, doi = {10.4230/LIPIcs.FSTTCS.2020.9}, annote = {Keywords: Parameterized algorithms, interval width, linear extension, one-sided crossing minimization, Kemeny rank aggregation, grouping by swapping} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We generalize the concept of synchronizing words for finite automata, which map all states of the automata to the same state, to deterministic visibly push-down automata. Here, a synchronizing word w does not only map all states to the same state but also fulfills some conditions on the stack content of each run after reading w. We consider three types of these stack constraints: after reading w, the stack (1) is empty in each run, (2) contains the same sequence of stack symbols in each run, or (3) contains an arbitrary sequence which is independent of the other runs. We show that in contrast to general deterministic push-down automata, it is decidable for deterministic visibly push-down automata whether there exists a synchronizing word with each of these stack constraints, more precisely, the problems are in EXPTIME. Under the constraint (1), the problem is even in P. For the sub-classes of deterministic very visibly push-down automata, the problem is in P for all three types of constraints. We further study variants of the synchronization problem where the number of turns in the stack height behavior caused by a synchronizing word is restricted, as well as the problem of synchronizing a variant of a sequential transducer, which shows some visibly behavior, by a word that synchronizes the states and produces the same output on all runs.

Henning Fernau and Petra Wolf. Synchronization of Deterministic Visibly Push-Down Automata. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fernau_et_al:LIPIcs.FSTTCS.2020.45, author = {Fernau, Henning and Wolf, Petra}, title = {{Synchronization of Deterministic Visibly Push-Down Automata}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {45:1--45:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.45}, URN = {urn:nbn:de:0030-drops-132861}, doi = {10.4230/LIPIcs.FSTTCS.2020.45}, annote = {Keywords: Synchronizing word, Deterministic visibly push-down automata, Deterministc finite atuomata, Finite-turn push-down automata, Sequential transducer, Computational complexity} }

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

The question if a deterministic finite automaton admits a software reset in the form of a so-called synchronizing word can be answered in polynomial time. In this paper, we extend this algorithmic question to deterministic automata beyond finite automata. We prove that the question of synchronizability becomes undecidable even when looking at deterministic one-counter automata. This is also true for another classical mild extension of regularity, namely that of deterministic one-turn push-down automata. However, when we combine both restrictions, we arrive at scenarios with a PSPACE-complete (and hence decidable) synchronizability problem. Likewise, we arrive at a decidable synchronizability problem for (partially) blind deterministic counter automata.
There are several interpretations of what synchronizability should mean for deterministic push-down automata. This is depending on the role of the stack: should it be empty on synchronization, should it be always the same or is it arbitrary? For the automata classes studied in this paper, the complexity or decidability status of the synchronizability problem is mostly independent of this technicality, but we also discuss one class of automata where this makes a difference.

Henning Fernau, Petra Wolf, and Tomoyuki Yamakami. Synchronizing Deterministic Push-Down Automata Can Be Really Hard. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fernau_et_al:LIPIcs.MFCS.2020.33, author = {Fernau, Henning and Wolf, Petra and Yamakami, Tomoyuki}, title = {{Synchronizing Deterministic Push-Down Automata Can Be Really Hard}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {33:1--33:15}, 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.2020.33}, URN = {urn:nbn:de:0030-drops-127008}, doi = {10.4230/LIPIcs.MFCS.2020.33}, annote = {Keywords: Synchronizing automaton, Reset sequence, Real-time deterministic push-down automaton, Finite-turn push-down automaton, Computability, Computational complexity} }

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

Many variations of synchronization of finite automata have been studied in the previous decades. Here, we suggest studying the question if synchronizing words exist that belong to some fixed constraint language, given by some partial finite automaton called constraint automaton. We show that this synchronization problem becomes PSPACE-complete even for some constraint automata with two states and a ternary alphabet. In addition, we characterize constraint automata with arbitrarily many states for which the constrained synchronization problem is polynomial-time solvable. We classify the complexity of the constrained synchronization problem for constraint automata with two states and two or three letters completely and lift those results to larger classes of finite automata.

Henning Fernau, Vladimir V. Gusev, Stefan Hoffmann, Markus Holzer, Mikhail V. Volkov, and Petra Wolf. Computational Complexity of Synchronization under Regular Constraints. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fernau_et_al:LIPIcs.MFCS.2019.63, author = {Fernau, Henning and Gusev, Vladimir V. and Hoffmann, Stefan and Holzer, Markus and Volkov, Mikhail V. and Wolf, Petra}, title = {{Computational Complexity of Synchronization under Regular Constraints}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {63:1--63:14}, 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.2019.63}, URN = {urn:nbn:de:0030-drops-110078}, doi = {10.4230/LIPIcs.MFCS.2019.63}, annote = {Keywords: Finite automata, synchronization, computational complexity} }

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**Published in:** Dagstuhl Reports, Volume 8, Issue 10 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar 18421 "Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative".
Enumeration problems require to list all wanted objects of the input as, e.g., particular subsets of the vertex or edge set of a given graph or particular satisfying assignments of logical expressions. Enumeration problems arise in a natural way in various fields of Computer Science, as, e.g., Artificial Intelligence and Data Mining, in Natural Sciences Engineering, Social Sciences, and Biology. The main challenge of the area of enumeration problems is that, contrary to decision, optimization and even counting problems, the output length of an enumeration problem is often exponential in the size of the input and cannot be neglected. This makes enumeration problems more challenging than many other types of algorithmic problems and demands development of specific techniques.
The principal goals of our Dagstuhl seminar were to increase the visibility of algorithmic enumeration within (Theoretical) Computer Science and to contribute to establishing it as an area of ``Algorithms and Complexity''. The seminar brought together researchers within the algorithms community, other fields of Computer Science and Computer Engineering, as well as researchers working on enumeration problems in other application areas, in particular Biology. The aim was to accelerate developments and discus new directions including algorithmic tools and hardness proofs.

Henning Fernau, Petr. A. Golovach, and Marie-France Sagot. Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative (Dagstuhl Seminar 18421). In Dagstuhl Reports, Volume 8, Issue 10, pp. 63-86, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{fernau_et_al:DagRep.8.10.63, author = {Fernau, Henning and Golovach, Petr. A. and Sagot, Marie-France}, title = {{Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative (Dagstuhl Seminar 18421)}}, pages = {63--86}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {8}, number = {10}, editor = {Fernau, Henning and Golovach, Petr. A. and Sagot, Marie-France}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.10.63}, URN = {urn:nbn:de:0030-drops-103483}, doi = {10.4230/DagRep.8.10.63}, annote = {Keywords: algorithms, input-sensitive enumeration, output-sensitive enumeration} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Unit square (grid) visibility graphs (USV and USGV, resp.) are described by axis-parallel visibility between unit squares placed (on integer grid coordinates) in the plane. We investigate combinatorial properties of these graph classes and the hardness of variants of the recognition problem, i.e., the problem of representing USGV with fixed visibilities within small area and, for USV, the general recognition problem.

Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L. Schmid, and Sue Whitesides. Combinatorial Properties and Recognition of Unit Square Visibility Graphs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 30:1-30:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{casel_et_al:LIPIcs.MFCS.2017.30, author = {Casel, Katrin and Fernau, Henning and Grigoriev, Alexander and Schmid, Markus L. and Whitesides, Sue}, title = {{Combinatorial Properties and Recognition of Unit Square Visibility Graphs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {30:1--30:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.30}, URN = {urn:nbn:de:0030-drops-80770}, doi = {10.4230/LIPIcs.MFCS.2017.30}, annote = {Keywords: Visibility graphs, visibility layout, NP-completeness, exact algorithms} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios however the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for k = 2 with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.

Faisal Abu-Khzam, Cristina Bazgan, Katrin Casel, and Henning Fernau. Building Clusters with Lower-Bounded Sizes. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abukhzam_et_al:LIPIcs.ISAAC.2016.4, author = {Abu-Khzam, Faisal and Bazgan, Cristina and Casel, Katrin and Fernau, Henning}, title = {{Building Clusters with Lower-Bounded Sizes}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {4:1--4:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.4}, URN = {urn:nbn:de:0030-drops-67742}, doi = {10.4230/LIPIcs.ISAAC.2016.4}, annote = {Keywords: Clustering, Approximation Algorithms, Complexity, Matching} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

It is shown that the shortest-grammar problem remains NP-complete if the alphabet is fixed and has a size of at least 24 (which settles an open question). On the other hand, this problem can be solved in polynomial-time, if the number of nonterminals is bounded, which is shown by encoding the problem as a problem on graphs with interval structure. Furthermore, we present an O(3n) exact exponential-time algorithm, based on dynamic programming. Similar results are also given for 1-level grammars, i.e., grammars for which only the start rule contains nonterminals on the right side (thus, investigating the impact of the "hierarchical depth" on the complexity of the shortest-grammar problem).

Katrin Casel, Henning Fernau, Serge Gaspers, Benjamin Gras, and Markus L. Schmid. On the Complexity of Grammar-Based Compression over Fixed Alphabets. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 122:1-122:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{casel_et_al:LIPIcs.ICALP.2016.122, author = {Casel, Katrin and Fernau, Henning and Gaspers, Serge and Gras, Benjamin and Schmid, Markus L.}, title = {{On the Complexity of Grammar-Based Compression over Fixed Alphabets}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {122:1--122:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.122}, URN = {urn:nbn:de:0030-drops-62570}, doi = {10.4230/LIPIcs.ICALP.2016.122}, annote = {Keywords: Grammar-Based Compression, Straight-Line Programs, NP-Completeness, Exact Exponential Time Algorithms} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

A pattern (i. e., a string of variables and terminals) maps to a word, if this is obtained by uniformly replacing the variables by terminal words; deciding this is NP-complete. We present efficient
algorithms\footnote{The computational model we use is the standard unit-cost RAM with logarithmic word size. Also, all logarithms appearing in our time complexity evaluations are in base 2.} that solve this problem for restricted classes of patterns. Furthermore, we show that it is NP-complete to decide, for a given number k and a word w, whether w can be factorised into k distinct factors; this shows that the injective version (i.e., different variables are replaced by different words) of the above matching problem is NP-complete even for very restricted cases.

Henning Fernau, Florin Manea, Robert Mercas, and Markus L. Schmid. Pattern Matching with Variables: Fast Algorithms and New Hardness Results. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 302-315, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{fernau_et_al:LIPIcs.STACS.2015.302, author = {Fernau, Henning and Manea, Florin and Mercas, Robert and Schmid, Markus L.}, title = {{Pattern Matching with Variables: Fast Algorithms and New Hardness Results}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {302--315}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.302}, URN = {urn:nbn:de:0030-drops-49220}, doi = {10.4230/LIPIcs.STACS.2015.302}, annote = {Keywords: combinatorial pattern matching, combinatorics on words, patterns with variables, \$\{cal NP\}\$-complete string problems} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Given a source string u and a target string w, to decide whether w can be obtained by applying a string morphism on u (i. e., uniformly
replacing the symbols in u by strings) constitutes an NP-complete
problem. For example, the target string w := baaba can be obtained from the source string u := aba, by replacing a and b in u by the strings ba and a, respectively. In this paper, we contribute to the recently started investigation of the computational complexity of the string morphism problem by studying it in the framework of parameterised complexity.

Henning Fernau, Markus L. Schmid, and Yngve Villanger. On the Parameterised Complexity of String Morphism Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 55-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{fernau_et_al:LIPIcs.FSTTCS.2013.55, author = {Fernau, Henning and Schmid, Markus L. and Villanger, Yngve}, title = {{On the Parameterised Complexity of String Morphism Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {55--66}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.55}, URN = {urn:nbn:de:0030-drops-43619}, doi = {10.4230/LIPIcs.FSTTCS.2013.55}, annote = {Keywords: String Problems, String Morphisms, Parameterised Complexity, Exponential Time Hypothesis, Pattern Languages} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

The {\sc $k$-Leaf Out-Branching} problem is to find an out-branching, that is a rooted oriented spanning tree, with at least $k$ leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the {\sc $k$-Leaf-Out-Branching} problem. We give the first polynomial kernel for {\sc Rooted $k$-Leaf-Out-Branching}, a variant of {\sc $k$-Leaf-Out-Branching} where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics.
For the {\sc $k$-Leaf-Out-Branching} problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a non-trivial fashion. However, our positive results for {\sc Rooted $k$-Leaf-Out-Branching} immediately imply that the seemingly intractable {\sc $k$-Leaf-Out-Branching} problem admits a data reduction to $n$ independent $O(k^3)$ kernels. These two results, tractability and intractability side by side, are the first ones separating {\it many-to-one kernelization} from {\it Turing kernelization}. This answers affirmatively an open problem regarding ``cheat kernelization'' raised by Mike Fellows and Jiong Guo independently.

Henning Fernau, Fedor V. Fomin, Daniel Lokshtanov, Daniel Raible, Saket Saurabh, and Yngve Villanger. Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 421-432, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{fernau_et_al:LIPIcs.STACS.2009.1843, author = {Fernau, Henning and Fomin, Fedor V. and Lokshtanov, Daniel and Raible, Daniel and Saurabh, Saket and Villanger, Yngve}, title = {{Kernel(s) for Problems with No Kernel: On Out-Trees with Many Leaves}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {421--432}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1843}, URN = {urn:nbn:de:0030-drops-18437}, doi = {10.4230/LIPIcs.STACS.2009.1843}, annote = {Keywords: Parameterized algorithms, Kernelization, Out-branching, Max-leaf, Lower bounds} }

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**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

We consider the construction of finite automata from their corresponding regular expressions by a series of digraph-transformations along the expression\'s structure.
Each intermediate graph represents an extended finite automaton accepting the same language. The character of our construction allows a fine-grained analysis of the
emerging automaton\'s size, eventually leading to an optimality result.

Stefan Gulan and Henning Fernau. An Optimal Construction of Finite Automata from Regular Expressions. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 211-222, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{gulan_et_al:LIPIcs.FSTTCS.2008.1754, author = {Gulan, Stefan and Fernau, Henning}, title = {{An Optimal Construction of Finite Automata from Regular Expressions}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {211--222}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1754}, URN = {urn:nbn:de:0030-drops-17544}, doi = {10.4230/LIPIcs.FSTTCS.2008.1754}, annote = {Keywords: Finite automata,regular expressions,descriptional complexity} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7281, Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs (2007)

One of the standard basic steps in drawing hierarchical graphs
is to invert some arcs of the given graph to make the graph acyclic.
We discuss exact and parameterized algorithms for this problem. In particular we examine a graph class called $(1,n)$-graphs, which contains cubic graphs. For both exact and parameterized algorithms we use a non-standard measure approach for the analysis. The analysis of the parameterized algorithm is of special interest, as it is not an amortized analysis modelled by 'finite states' but rather a 'top-down' amortized analysis. For $(1,n)$-graphs we achieve a running time of $Oh^*(1.1871^m)$ and $Oh^*(1.212^k)$, for cubic graphs $Oh^*(1.1798^m)$ and $Oh^*(1.201^k)$, respectively. As a by-product the trivial bound of $2^n$ for {sc Feedback Vertex Set} on planar directed graphs is broken.

Daniel Raible and Henning Fernau. Exact Elimination of Cycles in Graphs. In Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs. Dagstuhl Seminar Proceedings, Volume 7281, pp. 1-25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{raible_et_al:DagSemProc.07281.6, author = {Raible, Daniel and Fernau, Henning}, title = {{Exact Elimination of Cycles in Graphs}}, booktitle = {Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs}, pages = {1--25}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {7281}, editor = {Erik Demaine and Gregory Z. Gutin and Daniel Marx and Ulrike Stege}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07281.6}, URN = {urn:nbn:de:0030-drops-12353}, doi = {10.4230/DagSemProc.07281.6}, annote = {Keywords: Maximum Acyclic Subgraph, Feedback Arc Set, Amortized Analysis, Exact exponential algorthms} }