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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-dev.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)

The language P_n (P̃_n, respectively) consists of all words that are palindromes of length 2n (2n-1, respectively) over a fixed binary alphabet. We construct a regular expression that specifies P_n (P̃_n, respectively) of alphabetic width 4⋅ 2ⁿ-4 (3⋅ 2ⁿ-4, respectively) and show that this is optimal, that is, the expression has minimum alphabetic width among all expressions that describe P_n (P̃_n, respectively). To this end we give optimal expressions for the first k palindromes in lexicographic order of odd and even length, proving that the optimal bound is 2n+4(k-1)-2 S₂(k-1) in case of odd length and 2n+3(k-1)-2 S₂(k-1)-1 for even length, respectively. Here S₂(n) refers to the Hamming weight function, which denotes the number of ones in the binary expansion of the number n.

Hermann Gruber and Markus Holzer. Optimal Regular Expressions for Palindromes of Given Length. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gruber_et_al:LIPIcs.MFCS.2021.52, author = {Gruber, Hermann and Holzer, Markus}, title = {{Optimal Regular Expressions for Palindromes of Given Length}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {52:1--52:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.52}, URN = {urn:nbn:de:0030-drops-144921}, doi = {10.4230/LIPIcs.MFCS.2021.52}, annote = {Keywords: regular expression, descriptional complexity, lower bound, upper bound, recurrence, sum of digits} }

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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-dev.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:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

We continue our research on perversely awful randomized algorithms, which started nearly a decade ago. Based on the bogo-method we design a bogo-selection algorithm and variants thereof and analyse them with elementary methods. Moreover, practical experiments are performed.

Markus Holzer and Jan-Tobias Maurer. Selection Via the Bogo-Method - More on the Analysis of Perversely Awful Randomized Algorithms. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{holzer_et_al:LIPIcs.FUN.2018.23, author = {Holzer, Markus and Maurer, Jan-Tobias}, title = {{Selection Via the Bogo-Method - More on the Analysis of Perversely Awful Randomized Algorithms}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {23:1--23:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.23}, URN = {urn:nbn:de:0030-drops-88146}, doi = {10.4230/LIPIcs.FUN.2018.23}, annote = {Keywords: selection, bogo-method, combinatorial sums and series, inverse binomial coefficients, experimental result} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NP-complete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The two-action case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used to identify tractable cases of matrix elimination. We finally identify different classes of anonymous games where iterated dominance is in P and NP-complete, respectively.

Felix Brandt, Felix Fischer, and Markus Holzer. On Iterated Dominance, Matrix Elimination, and Matched Paths. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 107-118, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{brandt_et_al:LIPIcs.STACS.2010.2448, author = {Brandt, Felix and Fischer, Felix and Holzer, Markus}, title = {{On Iterated Dominance, Matrix Elimination, and Matched Paths}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {107--118}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2448}, URN = {urn:nbn:de:0030-drops-24485}, doi = {10.4230/LIPIcs.STACS.2010.2448}, annote = {Keywords: Algorithmic Game Theory, Computational Complexity, Iterated Dominance, Matching} }

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