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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

A recent result by Bodirsky and Guzmán-Pro gives a complexity dichotomy for the following class of computational problems, parametrized by a finite family F of finite tournaments: given an undirected graph, does there exist an orientation of the graph that avoids every tournament in F? One can see the edges of the input graphs as constraints imposing to find an orientation. In this paper, we consider a more general version of this problem where the constraints in the input are not necessarily about pairs of variables and impose local constraints on the global oriented graph to be found. Our main result is a complexity dichotomy for such problems, as well as a classification of such problems where the yes-instances have bounded treewidth duality. As a consequence, we obtain a streamlined proof of the result by Bodirsky and Guzmán-Pro using the theory of smooth approximations due to Mottet and Pinsker.

Zeno Bitter and Antoine Mottet. Generalized Completion Problems with Forbidden Tournaments. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bitter_et_al:LIPIcs.MFCS.2024.28, author = {Bitter, Zeno and Mottet, Antoine}, title = {{Generalized Completion Problems with Forbidden Tournaments}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {28:1--28:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.28}, URN = {urn:nbn:de:0030-drops-205844}, doi = {10.4230/LIPIcs.MFCS.2024.28}, annote = {Keywords: Tournaments, completion problems, constraint satisfaction problems, homogeneous structures, polymorphisms} }

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

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of some homogeneous hypergraphs. This forms a vast generalization of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongrácz (ICALP'16) on graph satisfiability.

Antoine Mottet, Tomáš Nagy, and Michael Pinsker. An Order out of Nowhere: A New Algorithm for Infinite-Domain {CSP}s. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 148:1-148:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mottet_et_al:LIPIcs.ICALP.2024.148, author = {Mottet, Antoine and Nagy, Tom\'{a}\v{s} and Pinsker, Michael}, title = {{An Order out of Nowhere: A New Algorithm for Infinite-Domain \{CSP\}s}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {148:1--148:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.148}, URN = {urn:nbn:de:0030-drops-202912}, doi = {10.4230/LIPIcs.ICALP.2024.148}, annote = {Keywords: Constraint Satisfaction Problems, Hypergraphs, Polymorphisms} }

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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Two particularly active branches of research in constraint satisfaction are the study of promise constraint satisfaction problems (PCSPs) with finite templates and the study of infinite-domain constraint satisfaction problems with ω-categorical templates. In this paper, we explore some connections between these two hitherto unrelated fields and describe a general approach to studying the complexity of PCSPs by constructing suitable infinite CSP templates. As a result, we obtain new characterizations of the power of various classes of algorithms for PCSPs, such as first-order logic, arc consistency reductions, and obtain new proofs of the characterizations of the power of the classical linear and affine relaxations for PCSPs.

Antoine Mottet. Promise and Infinite-Domain Constraint Satisfaction. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mottet:LIPIcs.CSL.2024.41, author = {Mottet, Antoine}, title = {{Promise and Infinite-Domain Constraint Satisfaction}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {41:1--41:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.41}, URN = {urn:nbn:de:0030-drops-196842}, doi = {10.4230/LIPIcs.CSL.2024.41}, annote = {Keywords: promise constraint satisfaction problems, polymorphisms, homogeneous structures, first-order logic} }

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

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Register automata are finite automata equipped with a finite set of registers ranging over the domain of some relational structure like (ℕ; =) or (ℚ; <). Register automata process words over the domain, and along a run of the automaton, the registers can store data from the input word for later comparisons. It is long known that the universality problem, i.e., the problem to decide whether a given register automaton accepts all words over the domain, is undecidable. Recently, we proved the problem to be decidable in 2-ExpSpace if the register automaton under study is over (ℕ; =) and unambiguous, i.e., every input word has at most one accepting run; this result was shortly after improved to 2-ExpTime by Barloy and Clemente. In this paper, we go one step further and prove that the problem is in ExpSpace, and in PSpace if the number of registers is fixed. Our proof is based on new techniques that additionally allow us to show that the problem is in PSpace for single-register automata over (ℚ; <). As a third technical contribution we prove that the problem is decidable (in ExpSpace) for a more expressive model of unambiguous register automata, where the registers can take values nondeterministically, if defined over (ℕ; =) and only one register is used.

Wojciech Czerwiński, Antoine Mottet, and Karin Quaas. New Techniques for Universality in Unambiguous Register Automata. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 129:1-129:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{czerwinski_et_al:LIPIcs.ICALP.2021.129, author = {Czerwi\'{n}ski, Wojciech and Mottet, Antoine and Quaas, Karin}, title = {{New Techniques for Universality in Unambiguous Register Automata}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {129:1--129:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.129}, URN = {urn:nbn:de:0030-drops-141983}, doi = {10.4230/LIPIcs.ICALP.2021.129}, annote = {Keywords: Register Automata, Data Languages, Unambiguity, Unambiguous, Universality, Containment, Language Inclusion, Equivalence} }

Document

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

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ≥ 3) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al.. In particular, the bounded width hierarchy collapses in those cases as well.

Antoine Mottet, Tomáš Nagy, Michael Pinsker, and Michał Wrona. Smooth Approximations and Relational Width Collapses. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 138:1-138:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{mottet_et_al:LIPIcs.ICALP.2021.138, author = {Mottet, Antoine and Nagy, Tom\'{a}\v{s} and Pinsker, Michael and Wrona, Micha{\l}}, title = {{Smooth Approximations and Relational Width Collapses}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {138:1--138:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.138}, URN = {urn:nbn:de:0030-drops-142075}, doi = {10.4230/LIPIcs.ICALP.2021.138}, annote = {Keywords: local consistency, bounded width, constraint satisfaction problems, polymorphisms, smooth approximations} }

Document

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

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

We produce a class of ω-categorical structures with finite signature by applying a model-theoretic construction - a refinement of an encoding due to Hrushosvki - to ω-categorical structures in a possibly infinite signature. We show that the encoded structures retain desirable algebraic properties of the original structures, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity. This method allows us to systematically generate ω-categorical templates whose CSPs are complete for a variety of complexity classes of arbitrarily high complexity, and ω-categorical templates that show that membership in any given complexity class cannot be expressed by a set of identities on the polymorphisms. It moreover enables us to prove that recent results about the relevance of topology on polymorphism clones of ω-categorical structures also apply for CSP templates, i.e., structures in a finite language. Finally, we obtain a concrete algebraic criterion which could constitute a description of the delineation between tractability and NP-hardness in the dichotomy conjecture for first-order reducts of finitely bounded homogeneous structures.

Pierre Gillibert, Julius Jonušas, Michael Kompatscher, Antoine Mottet, and Michael Pinsker. Hrushovski’s Encoding and ω-Categorical CSP Monsters. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 131:1-131:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gillibert_et_al:LIPIcs.ICALP.2020.131, author = {Gillibert, Pierre and Jonu\v{s}as, Julius and Kompatscher, Michael and Mottet, Antoine and Pinsker, Michael}, title = {{Hrushovski’s Encoding and \omega-Categorical CSP Monsters}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {131:1--131:17}, 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.2020.131}, URN = {urn:nbn:de:0030-drops-125387}, doi = {10.4230/LIPIcs.ICALP.2020.131}, annote = {Keywords: Constraint satisfaction problem, complexity, polymorphism, pointwise convergence topology, height 1 identity, \omega-categoricity, orbit growth} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We investigate the complexity of the containment problem "Does L(A)subseteq L(B) hold?", where B is an unambiguous register automaton and A is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when B is restricted to have a fixed number of registers.

Antoine Mottet and Karin Quaas. The Containment Problem for Unambiguous Register Automata. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mottet_et_al:LIPIcs.STACS.2019.53, author = {Mottet, Antoine and Quaas, Karin}, title = {{The Containment Problem for Unambiguous Register Automata}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {53:1--53:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.53}, URN = {urn:nbn:de:0030-drops-102926}, doi = {10.4230/LIPIcs.STACS.2019.53}, annote = {Keywords: Data words, Register automata, Unambiguous Automata, Containment Problem, Language Inclusion Problem} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We study the Constraint Satisfaction Problem CSP( A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete.

Manuel Bodirsky, Barnaby Martin, Marcello Mamino, and Antoine Mottet. The Complexity of Disjunctive Linear Diophantine Constraints. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bodirsky_et_al:LIPIcs.MFCS.2018.33, author = {Bodirsky, Manuel and Martin, Barnaby and Mamino, Marcello and Mottet, Antoine}, title = {{The Complexity of Disjunctive Linear Diophantine Constraints}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {33:1--33:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.33}, URN = {urn:nbn:de:0030-drops-96150}, doi = {10.4230/LIPIcs.MFCS.2018.33}, annote = {Keywords: Constraint Satisfaction, Presburger Arithmetic, Computational Complexity} }

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