Document

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

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We show that the problem of deciding for a given finite relation algebra A whether the network satisfaction problem for A can be solved by the k-consistency procedure, for some k ∈ ℕ, is undecidable. For the important class of finite relation algebras A with a normal representation, however, the decidability of this problem remains open. We show that if A is symmetric and has a flexible atom, then the question whether NSP(A) can be solved by k-consistency, for some k ∈ ℕ, is decidable (even in polynomial time in the number of atoms of A). This result follows from a more general sufficient condition for the correctness of the k-consistency procedure for finite symmetric relation algebras. In our proof we make use of a result of Alexandr Kazda about finite binary conservative structures.

Manuel Bodirsky and Simon Knäuer. Network Satisfaction Problems Solved by k-Consistency. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 116:1-116:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2023.116, author = {Bodirsky, Manuel and Kn\"{a}uer, Simon}, title = {{Network Satisfaction Problems Solved by k-Consistency}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {116:1--116:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.116}, URN = {urn:nbn:de:0030-drops-181680}, doi = {10.4230/LIPIcs.ICALP.2023.116}, annote = {Keywords: Constraint Satisfaction, Computational Complexity, Relation Algebras, Network Satisfaction, Qualitative Reasoning, k-Consistency, Datalog} }

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 characterise the sentences in Monadic Second-order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class C of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all 𝓁,k ∈ , there exists a canonical Datalog program Π of width (𝓁,k), that is, a Datalog program of width (𝓁,k) which is sound for C (i.e., Π only derives the goal predicate on a finite structure 𝔄 if 𝔄 ∈ C) and with the property that Π derives the goal predicate whenever some Datalog program of width (𝓁,k) which is sound for C derives the goal predicate. The same characterisations also hold for Guarded Second-order Logic (GSO), which properly extends MSO. To prove our results, we show that every class C in GSO whose complement is closed under homomorphisms is a finite union of constraint satisfaction problems (CSPs) of ω-categorical structures.

Manuel Bodirsky, Simon Knäuer, and Sebastian Rudolph. Datalog-Expressibility for Monadic and Guarded Second-Order Logic. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 120:1-120:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2021.120, author = {Bodirsky, Manuel and Kn\"{a}uer, Simon and Rudolph, Sebastian}, title = {{Datalog-Expressibility for Monadic and Guarded Second-Order Logic}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {120:1--120:17}, 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.2021.120}, URN = {urn:nbn:de:0030-drops-141897}, doi = {10.4230/LIPIcs.ICALP.2021.120}, annote = {Keywords: Monadic Second-order Logic, Guarded Second-order Logic, Datalog, constraint satisfaction, homomorphism-closed, conjunctive query, primitive positive formula, pebble game, \omega-categoricity} }

Document

**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear homogeneous cost functions. We remark that in this paper the infinite domain will always be the set of rational numbers. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP.

Manuel Bodirsky, Marcello Mamino, and Caterina Viola. Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bodirsky_et_al:LIPIcs.CSL.2018.12, author = {Bodirsky, Manuel and Mamino, Marcello and Viola, Caterina}, title = {{Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.12}, URN = {urn:nbn:de:0030-drops-96792}, doi = {10.4230/LIPIcs.CSL.2018.12}, annote = {Keywords: Valued constraint satisfaction problems, Piecewise linear functions, Submodular functions, Semilinear, Constraint satisfaction, Optimisation, Model Theory} }

Document

**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} }

Document

**Published in:** Dagstuhl Follow-Ups, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability (2017)

We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert's tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra.

Manuel Bodirsky and Marcello Mamino. Constraint Satisfaction Problems over Numeric Domains. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 79-111, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{bodirsky_et_al:DFU.Vol7.15301.79, author = {Bodirsky, Manuel and Mamino, Marcello}, title = {{Constraint Satisfaction Problems over Numeric Domains}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {79--111}, series = {Dagstuhl Follow-Ups}, ISBN = {978-3-95977-003-3}, ISSN = {1868-8977}, year = {2017}, volume = {7}, editor = {Krokhin, Andrei and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DFU.Vol7.15301.79}, URN = {urn:nbn:de:0030-drops-69580}, doi = {10.4230/DFU.Vol7.15301.79}, annote = {Keywords: Constraint satisfaction problems, Numerical domains} }

Document

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

For n >= 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Gamma with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Gamma is either in P or is NP-complete.
We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation.
Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.

Manuel Bodirsky, Barnaby Martin, Michael Pinsker, and András Pongrácz. Constraint Satisfaction Problems for Reducts of Homogeneous Graphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 119:1-119:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2016.119, author = {Bodirsky, Manuel and Martin, Barnaby and Pinsker, Michael and Pongr\'{a}cz, Andr\'{a}s}, title = {{Constraint Satisfaction Problems for Reducts of Homogeneous Graphs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {119:1--119:14}, 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.2016.119}, URN = {urn:nbn:de:0030-drops-62543}, doi = {10.4230/LIPIcs.ICALP.2016.119}, annote = {Keywords: Constraint Satisfaction, Homogeneous Graphs, Computational Complexity, Universal Algebra, Ramsey Theory} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains for example the rooted triple consistency problem, forbidden subtree problems, the quartet consistency problem, and many other problems studied in the bioinformatics literature. The studied problems can be described as constraint satisfaction problems where the constraints have a first-order definition over the rooted triple relation. We show that every such phylogeny problem can be solved in polynomial time or is NP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithm of Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithm repeatedly solves linear equation systems to construct a solution in polynomial time. We then show that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Our classification establishes a dichotomy for a large class of infinite structures that we believe is of independent interest in universal algebra, model theory, and topology. The proof of our main result combines results and techniques from various research areas: a recent classification of the model-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb’s Ramsey theorem for rooted trees, and universal algebra.

Manuel Bodirsky, Peter Jonsson, and Trung Van Pham. The Complexity of Phylogeny Constraint Satisfaction. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 20:1-20:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bodirsky_et_al:LIPIcs.STACS.2016.20, author = {Bodirsky, Manuel and Jonsson, Peter and Van Pham, Trung}, title = {{The Complexity of Phylogeny Constraint Satisfaction}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {20:1--20:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.20}, URN = {urn:nbn:de:0030-drops-57218}, doi = {10.4230/LIPIcs.STACS.2016.20}, annote = {Keywords: constraint satisfaction problems, computational complexity, phylogenetic reconstruction, ramsey theory, model theory} }

Document

Invited Talk

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

The tractability conjecture for constraint satisfaction problems (CSPs)
describes the constraint languages over a finite domain whose CSP can be solved in polynomial-time. The precise formulation of the conjecture
uses basic notions from universal algebra. In this talk, we give a short introduction to the universal-algebraic approach to the study of the complexity of CSPs. Finally, we discuss attempts to generalise the tractability conjecture to large classes of constraint languages over infinite domains, in particular for constraint languages that arise in qualitative temporal and spatial reasoning.

Manuel Bodirsky. The Complexity of Constraint Satisfaction Problems (Invited Talk). In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 2-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bodirsky:LIPIcs.STACS.2015.2, author = {Bodirsky, Manuel}, title = {{The Complexity of Constraint Satisfaction Problems}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {2--9}, 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.2015.2}, URN = {urn:nbn:de:0030-drops-49567}, doi = {10.4230/LIPIcs.STACS.2015.2}, annote = {Keywords: constraint satisfaction, universal algebra, model theory, clones, temporal and spatial reasoning} }

Document

**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

The following result for finite structures Gamma has been conjectured to hold for all countably infinite omega-categorical structures Gamma: either the model-complete core Delta of Gamma has an expansion by finitely many constants such that the pseudovariety generated by its polymorphism algebra contains a two-element algebra all of whose operations are projections, or there is a homomorphism f from Delta^k to Delta, for some finite k, and an automorphism alpha of Delta satisfying f(x1,...,xk) = alpha(f(x2,...,xk,x1)). This conjecture has been confirmed for all infinite structures Gamma that have a first-order definition over (Q;<), and for all structures that are definable over the random graph. In this paper, we verify the conjecture for all structures that are definable over an equivalence relation with a countably infinite number of countably infinite classes.
Our result implies a complexity dichotomy (into NP-complete and P) for a family of constraint satisfaction problems (CSPs) which we call equivalence constraint satisfaction problems. The classification for equivalence CSPs can also be seen as a first step towards a classification of the CSPs for all relational structures that are first-order definable over Allen's interval algebra, a well-known constraint calculus in temporal reasoning.

Manuel Bodirsky and Michal Wrona. Equivalence Constraint Satisfaction Problems. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 122-136, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{bodirsky_et_al:LIPIcs.CSL.2012.122, author = {Bodirsky, Manuel and Wrona, Michal}, title = {{Equivalence Constraint Satisfaction Problems}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {122--136}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.122}, URN = {urn:nbn:de:0030-drops-36689}, doi = {10.4230/LIPIcs.CSL.2012.122}, annote = {Keywords: Constraint satisfaction problems, universal algebra, model theory, Ram- sey theory, temporal reasoning, computational complexity} }