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**Published in:** LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary and n the number of states of the finite automata considered. The following main results are obtained:
1) Equality and inclusion of NFAs can be decided within time 2^O((n log n)^{1/3}); previous upper bound 2^O((n log n)^{1/2}) was by Chrobak (1986) via DFA conversion.
2) The state complexity of operations of UFAs (unambiguous finite automata) increases for complementation and union at most by quasipolynomial; however, for concatenation of two n-state UFAs, the worst case is an UFA of at least 2^Ω(n^{1/6}) states. Previously the upper bounds for complementation and union were exponential-type and this lower bound for concatenation is new.

Wojciech Czerwiński, Maciej Dębski, Tomasz Gogasz, Gordon Hoi, Sanjay Jain, Michał Skrzypczak, Frank Stephan, and Christopher Tan. Languages Given by Finite Automata over the Unary Alphabet. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{czerwinski_et_al:LIPIcs.FSTTCS.2023.22, author = {Czerwi\'{n}ski, Wojciech and D\k{e}bski, Maciej and Gogasz, Tomasz and Hoi, Gordon and Jain, Sanjay and Skrzypczak, Micha{\l} and Stephan, Frank and Tan, Christopher}, title = {{Languages Given by Finite Automata over the Unary Alphabet}}, booktitle = {43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)}, pages = {22:1--22:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-304-1}, ISSN = {1868-8969}, year = {2023}, volume = {284}, editor = {Bouyer, Patricia and Srinivasan, Srikanth}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.22}, URN = {urn:nbn:de:0030-drops-193959}, doi = {10.4230/LIPIcs.FSTTCS.2023.22}, annote = {Keywords: Nondeterministic Finite Automata, Unambiguous Finite Automata, Upper Bounds on Runtime, Conditional Lower Bounds, Languages over the Unary Alphabet} }

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

X3SAT is the problem of whether one can satisfy a given set of clauses with up to three literals such that in every clause, exactly one literal is true and the others are false. A related question is to determine the maximal Hamming distance between two solutions of the instance. Dahllöf provided an algorithm for Maximum Hamming Distance XSAT, which is more complicated than the same problem for X3SAT, with a runtime of O(1.8348^n); Fu, Zhou and Yin considered Maximum Hamming Distance for X3SAT and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an algorithm in O(1.3298^n) time to solve the Max Hamming Distance X3SAT problem; the algorithm actually counts for each k the number of pairs of solutions which have Hamming Distance k.

Gordon Hoi, Sanjay Jain, and Frank Stephan. A Fast Exponential Time Algorithm for Max Hamming Distance X3SAT. 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. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hoi_et_al:LIPIcs.FSTTCS.2019.17, author = {Hoi, Gordon and Jain, Sanjay and Stephan, Frank}, title = {{A Fast Exponential Time Algorithm for Max Hamming Distance X3SAT}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {17:1--17:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.17}, URN = {urn:nbn:de:0030-drops-115799}, doi = {10.4230/LIPIcs.FSTTCS.2019.17}, annote = {Keywords: X3SAT Problem, Maximum Hamming Distance of Solutions, Exponential Time Algorithms, DPLL Algorithms} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

XSAT is defined as the following: Given a propositional formula in conjunctive normal form, can one find an assignment to variables such that there is exactly only 1 literal that is true in every clause, while the other literals are false. The decision problem XSAT is known to be NP-complete. Crescenzi and Rossi [Pierluigi Crescenzi and Gianluca Rossi, 2002] introduced the variant where one searches for a pair of two solutions of an X3SAT instance with maximal Hamming Distance among them, that is, one wants to identify the largest number k such that there are two solutions of the instance with Hamming Distance k. Dahllöf [Vilhelm Dahllöf, 2005; Vilhelm Dahllöf, 2006] provided an algorithm using branch and bound method for Max Hamming Distance XSAT in O(1.8348^n); Fu, Zhou and Yin [Linlu Fu and Minghao Yin, 2012] worked on a more specific problem, the Max Hamming Distance X3SAT, and found for this problem an algorithm with runtime O(1.6760^n). In this paper, we propose an exact exponential algorithm to solve the Max Hamming Distance XSAT problem in O(1.4983^n) time. Like all of them, we will use the branch and bound technique alongside a newly defined measure to improve the analysis of the algorithm.

Gordon Hoi and Frank Stephan. Measure and Conquer for Max Hamming Distance XSAT. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hoi_et_al:LIPIcs.ISAAC.2019.15, author = {Hoi, Gordon and Stephan, Frank}, title = {{Measure and Conquer for Max Hamming Distance XSAT}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.15}, URN = {urn:nbn:de:0030-drops-115119}, doi = {10.4230/LIPIcs.ISAAC.2019.15}, annote = {Keywords: XSAT, Measure and Conquer, DPLL, Exponential Time Algorithms} }

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