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**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} }

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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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**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} }

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