Found 2 Possible Name Variants:

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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 give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.

Alberto Larrauri and Stanislav Živný. Solving Promise Equations over Monoids and Groups. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 146:1-146:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{larrauri_et_al:LIPIcs.ICALP.2024.146, author = {Larrauri, Alberto and \v{Z}ivn\'{y}, Stanislav}, title = {{Solving Promise Equations over Monoids and Groups}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {146:1--146:18}, 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.2024.146}, URN = {urn:nbn:de:0030-drops-202893}, doi = {10.4230/LIPIcs.ICALP.2024.146}, annote = {Keywords: constraint satisfaction, promise constraint satisfaction, equations, minions} }

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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

General factors are a generalization of matchings. Given a graph G with a set π(v) of feasible degrees, called a degree constraint, for each vertex v of G, the general factor problem is to find a (spanning) subgraph F of G such that deg_F(v) ∈ π(v) for every v of G. When all degree constraints are symmetric Δ-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. Strongly polynomial-time algorithms are only known for weighted general factor problems that are reducible to the weighted matching problem by gadget constructions.
In this paper, we present a strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions. As an application, we obtain a strongly polynomial-time algorithm for the terminal backup problem by reducing it to the weighted general factor problem.

Shuai Shao and Stanislav Živný. A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{shao_et_al:LIPIcs.ISAAC.2023.57, author = {Shao, Shuai and \v{Z}ivn\'{y}, Stanislav}, title = {{A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {57:1--57:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.57}, URN = {urn:nbn:de:0030-drops-193597}, doi = {10.4230/LIPIcs.ISAAC.2023.57}, annote = {Keywords: matchings, factors, edge constraint satisfaction problems, terminal backup problem, delta matroids} }

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**Published in:** Dagstuhl Reports, Volume 12, Issue 5 (2022)

Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes. CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena. For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter tractable, or definable in a weak logic). In the last 15 years, research activity in this area has significantly intensified and hugely impressive progress was made. The Dagstuhl Seminar 22201 "The Constraint Satisfaction Problem: Complexity and Approximability" was aimed at bringing together researchers using all the different techniques in the study of the CSP so that they can share their insights obtained during the past four years. This report documents the material presented during the course of the seminar.

Martin Grohe, Venkatesan Guruswami, Dániel Marx, and Stanislav Živný. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201). In Dagstuhl Reports, Volume 12, Issue 5, pp. 112-130, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{grohe_et_al:DagRep.12.5.112, author = {Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav}, title = {{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201)}}, pages = {112--130}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {12}, number = {5}, editor = {Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.5.112}, URN = {urn:nbn:de:0030-drops-174453}, doi = {10.4230/DagRep.12.5.112}, annote = {Keywords: Constraint satisfaction problem (CSP); Computational complexity; Hardness of approximation; Universal algebra; Semidefinite programming} }

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

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, …, k} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results.
First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with k = O(√{nlog log n}/log n), where n is the number of vertices of the input hypergraph. This is established by building on ideas from algorithms designed for approximate graph colourings.
Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO 3-colouring for every constant uniformity r ≥ 5. In fact, we determine the precise relationship of polymorphism minions for all uniformities r ≥ 3, which reveals a key difference between r = 3,4 and r ≥ 5 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO (k+1)-colouring for LO k-colourable r-uniform hypergraphs for k ≥ 2 and r ≥ 5.

Tamio-Vesa Nakajima and Stanislav Živný. Linearly Ordered Colourings of Hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 128:1-128:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{nakajima_et_al:LIPIcs.ICALP.2022.128, author = {Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav}, title = {{Linearly Ordered Colourings of Hypergraphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {128:1--128:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.128}, URN = {urn:nbn:de:0030-drops-164692}, doi = {10.4230/LIPIcs.ICALP.2022.128}, annote = {Keywords: hypegraph colourings, promise constraint satisfaction, PCSP, polymorphisms, minions, algebraic approach} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{larose_et_al:LIPIcs.ESA.2021.58, author = {Larose, Beno\^{i}t and Markovi\'{c}, Petar and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani and \v{Z}ivn\'{y}, Stanislav}, title = {{QCSP on Reflexive Tournaments}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {58:1--58:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.58}, URN = {urn:nbn:de:0030-drops-146392}, doi = {10.4230/LIPIcs.ESA.2021.58}, annote = {Keywords: computational complexity, algorithmic graph theory, quantified constraints, universal algebra, constraint satisfaction} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20].
Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs.
As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P:{0,1}^k → {0,1} of a fixed arity k, we show that CSP(P)} admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P)} admits an additive sparsifier for any predicate P:D^k → {0,1} of a fixed arity k on an arbitrary finite domain D.

Eden Pelleg and Stanislav Živný. Additive Sparsification of CSPs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pelleg_et_al:LIPIcs.ESA.2021.75, author = {Pelleg, Eden and \v{Z}ivn\'{y}, Stanislav}, title = {{Additive Sparsification of CSPs}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {75:1--75:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.75}, URN = {urn:nbn:de:0030-drops-146562}, doi = {10.4230/LIPIcs.ESA.2021.75}, annote = {Keywords: additive sparsification, graphs, hypergraphs, minimum cuts} }

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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)

The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs.
1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SODA'18] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (𝟏-in-𝟑,NAE) is tractable.
We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (𝐭-in-𝐤, NAE) by either adding tuples to 𝐭-in-𝐤 or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [SODA'20]. For the latter, we classify all templates as either tractable or NP-hard.

Alex Brandts and Stanislav Živný. Beyond PCSP(1-in-3, NAE). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 121:1-121:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brandts_et_al:LIPIcs.ICALP.2021.121, author = {Brandts, Alex and \v{Z}ivn\'{y}, Stanislav}, title = {{Beyond PCSP(1-in-3, NAE)}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {121:1--121:14}, 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.2021.121}, URN = {urn:nbn:de:0030-drops-141902}, doi = {10.4230/LIPIcs.ICALP.2021.121}, annote = {Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).

Caterina Viola and Stanislav Živný. The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{viola_et_al:LIPIcs.MFCS.2020.85, author = {Viola, Caterina and \v{Z}ivn\'{y}, Stanislav}, title = {{The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {85:1--85:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.85}, URN = {urn:nbn:de:0030-drops-127566}, doi = {10.4230/LIPIcs.MFCS.2020.85}, annote = {Keywords: promise constraint satisfaction, valued constraint satisfaction, convex relaxations, polymorphisms} }

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Track A: Algorithms, Complexity and Games

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

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and Håstad [FOCS'14/SICOMP'17] proved a result known as "(2+ε)-SAT is NP-hard". They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains.

Alex Brandts, Marcin Wrochna, and Stanislav Živný. The Complexity of Promise SAT on Non-Boolean Domains. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brandts_et_al:LIPIcs.ICALP.2020.17, author = {Brandts, Alex and Wrochna, Marcin and \v{Z}ivn\'{y}, Stanislav}, title = {{The Complexity of Promise SAT on Non-Boolean Domains}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {17:1--17:13}, 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.2020.17}, URN = {urn:nbn:de:0030-drops-124241}, doi = {10.4230/LIPIcs.ICALP.2020.17}, annote = {Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach, label cover} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04].
We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors.

Andrei A. Bulatov and Stanislav Živný. Approximate Counting CSP Seen from the Other Side. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bulatov_et_al:LIPIcs.MFCS.2019.60, author = {Bulatov, Andrei A. and \v{Z}ivn\'{y}, Stanislav}, title = {{Approximate Counting CSP Seen from the Other Side}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {60:1--60:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.60}, URN = {urn:nbn:de:0030-drops-110041}, doi = {10.4230/LIPIcs.MFCS.2019.60}, annote = {Keywords: constraint satisfaction, approximate counting, homomorphisms} }

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

A cut epsilon-sparsifier of a weighted graph G is a re-weighted subgraph of G of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of epsilon. Since their introduction by Benczúr and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.

Silvia Butti and Stanislav Živný. Sparsification of Binary CSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 17:1-17:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{butti_et_al:LIPIcs.STACS.2019.17, author = {Butti, Silvia and \v{Z}ivn\'{y}, Stanislav}, title = {{Sparsification of Binary CSPs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {17:1--17:8}, 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.2019.17}, URN = {urn:nbn:de:0030-drops-102564}, doi = {10.4230/LIPIcs.STACS.2019.17}, annote = {Keywords: constraint satisfaction problems, minimum cuts, sparsification} }

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

Fulla, Uppman, and Živný [ACM ToCT'18] established a dichotomy theorem for Boolean surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs on two-element domains in which both labels have to be used in a solution. This result, in addition to identifying the complexity frontier, features the discovery of a new non-trivial tractable case (called EDS) that does not appear in the non-surjective setting.
In this work, we go beyond Boolean domains. As our main result, we introduce a generalisation of EDS to arbitrary finite domains called SEDS (similar to EDS) and establish a conditional complexity classification of SEDS VCSPs based on a reduction to smaller domains. This gives a complete classification of SEDS VCSPs on three-element domains. The basis of our tractability result is a natural generalisation of the Min-Cut problem, in which only solutions of certain size (given by a lower and upper bound) are permitted. We show that all near-optimal solutions to this problem can be enumerated in polynomial time, which might be of independent interest.

Gregor Matl and Stanislav Živný. Beyond Boolean Surjective VCSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{matl_et_al:LIPIcs.STACS.2019.52, author = {Matl, Gregor and \v{Z}ivn\'{y}, Stanislav}, title = {{Beyond Boolean Surjective VCSPs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {52:1--52:15}, 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.2019.52}, URN = {urn:nbn:de:0030-drops-102911}, doi = {10.4230/LIPIcs.STACS.2019.52}, annote = {Keywords: constraint satisfaction problems, valued constraint satisfaction, surjective constraint satisfaction, graph cuts} }

Document

**Published in:** Dagstuhl Reports, Volume 8, Issue 6 (2019)

Constraint satisfaction has always played a central role in computational
complexity theory; appropriate versions of CSPs are classical complete
problems for most standard complexity classes. CSPs constitute a very rich and
yet sufficiently manageable class of problems to give a good perspective on
general computational phenomena. For instance, they help to understand which
mathematical properties make a computational problem tractable (in a wide
sense, e.g., polynomial-time solvable, non-trivially approximable,
fixed-parameter tractable, or definable in a weak logic). In the last decade,
research activity in this area has significantly intensified and hugely
impressive progress was made.
The Dagstuhl Seminar 18231 "The Constraint Satisfaction Problem: Complexity and
Approximability" was aimed at bringing together researchers using all the
different techniques in the study of the CSP so that they can share their
insights obtained during the past three years. This report documents the
material presented during the course of the seminar.

Martin Grohe, Venkatesan Guruswami, and Stanislav Zivny. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231). In Dagstuhl Reports, Volume 8, Issue 6, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{grohe_et_al:DagRep.8.6.1, author = {Grohe, Martin and Guruswami, Venkatesan and Zivny, Stanislav}, title = {{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)}}, pages = {1--18}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {8}, number = {6}, editor = {Grohe, Martin and Guruswami, Venkatesan and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.6.1}, URN = {urn:nbn:de:0030-drops-100251}, doi = {10.4230/DagRep.8.6.1}, annote = {Keywords: Constraint satisfaction problem (CSP); Computational complexity; CSP dichotomy conjecture; Hardness of approximation; Unique games conjecture; Parameterised complexity; Descriptive complexity; Universal algebra; Logic; Semidefinite programming} }

Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

Clément Carbonnel, David A. Cohen, Martin C. Cooper, and Stanislav Zivny. On Singleton Arc Consistency for CSPs Defined by Monotone Patterns. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carbonnel_et_al:LIPIcs.STACS.2018.19, author = {Carbonnel, Cl\'{e}ment and Cohen, David A. and Cooper, Martin C. and Zivny, Stanislav}, title = {{On Singleton Arc Consistency for CSPs Defined by Monotone Patterns}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.19}, URN = {urn:nbn:de:0030-drops-84876}, doi = {10.4230/LIPIcs.STACS.2018.19}, annote = {Keywords: constraint satisfaction problems, forbidden patterns, singleton arc consistency} }

Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions.An important line of VCSP research is to investigate what functions can be solved in polynomial time.
Cooper-Zivny classified the tractability of binary VCSP instances according to the concept of "triangle,"
and showed that the only interesting tractable case is the one induced by the joint winner property (JWP).
Recently, Iwamasa-Murota-Zivny made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given.
In this paper,
we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be solved in polynomial time via an M-convex intersection algorithm?
We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case.
Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.

Hiroshi Hirai, Yuni Iwamasa, Kazuo Murota, and Stanislav Zivny. Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hirai_et_al:LIPIcs.STACS.2018.39, author = {Hirai, Hiroshi and Iwamasa, Yuni and Murota, Kazuo and Zivny, Stanislav}, title = {{Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.39}, URN = {urn:nbn:de:0030-drops-85042}, doi = {10.4230/LIPIcs.STACS.2018.39}, annote = {Keywords: valued constraint satisfaction problems, discrete convex analysis, M-convexity} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with the objective function given as a sum of fixed-arity functions; the values are rational numbers or infinity.
In Boolean surjective VCSPs variables take on labels from D={0,1} and an optimal assignment is required to use both labels from D. A classic example is the global min-cut problem in graphs. Building on the work of Uppman, we establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs. The newly discovered tractable case has an interesting structure related to projections of downsets and upsets. Our work generalises the dichotomy for {0,infinity}-valued constraint languages corresponding to CSPs) obtained by Creignou and Hebrard, and the dichotomy for {0,1}-valued constraint languages (corresponding to Min-CSPs) obtained by Uppman.

Peter Fulla and Stanislav Zivny. The Complexity of Boolean Surjective General-Valued CSPs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fulla_et_al:LIPIcs.MFCS.2017.4, author = {Fulla, Peter and Zivny, Stanislav}, title = {{The Complexity of Boolean Surjective General-Valued CSPs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.4}, URN = {urn:nbn:de:0030-drops-80623}, doi = {10.4230/LIPIcs.MFCS.2017.4}, annote = {Keywords: constraint satisfaction problems, surjective CSP, valued CSP, min-cut, polymorphisms, multimorphisms} }

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Complete Volume

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

DFU, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability, Complete Volume

The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Collection{DFU.Vol7.15301, title = {{DFU, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability, Complete Volume}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, 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}, URN = {urn:nbn:de:0030-drops-69752}, doi = {10.4230/DFU.Vol7.15301}, annote = {Keywords: Nonnumerical Algorithms and Problems} }

Document

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

Front Matter, Table of Contents, Preface, List of Authors

The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{krokhin_et_al:DFU.Vol7.15301.i, author = {Krokhin, Andrei and Zivny, Stanislav}, title = {{Front Matter, Table of Contents, Preface, List of Authors}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {0:i--0:xii}, 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.i}, URN = {urn:nbn:de:0030-drops-69702}, doi = {10.4230/DFU.Vol7.15301.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, List of Authors} }

Document

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

We present a survey of complexity results for hybrid constraint satisfaction problems (CSPs) and valued constraint satisfaction problems (VCSPs). These are classes of (V)CSPs defined by restrictions that are not exclusively language-based or structure-based.

Martin C. Cooper and Stanislav Zivny. Hybrid Tractable Classes of Constraint Problems. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 113-135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{cooper_et_al:DFU.Vol7.15301.113, author = {Cooper, Martin C. and Zivny, Stanislav}, title = {{Hybrid Tractable Classes of Constraint Problems}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {113--135}, 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.Vol7.15301.113}, URN = {urn:nbn:de:0030-drops-69616}, doi = {10.4230/DFU.Vol7.15301.113}, annote = {Keywords: Constraint satisfaction problems, Optimisation, Tractability} }

Document

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

We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems (VCSPs). We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape.

Andrei Krokhin and Stanislav Zivny. The Complexity of Valued CSPs. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 233-266, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{krokhin_et_al:DFU.Vol7.15301.233, author = {Krokhin, Andrei and Zivny, Stanislav}, title = {{The Complexity of Valued CSPs}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {233--266}, 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.Vol7.15301.233}, URN = {urn:nbn:de:0030-drops-69665}, doi = {10.4230/DFU.Vol7.15301.233}, annote = {Keywords: Constraint satisfaction problems, Optimisation, Tractability} }

Document

**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We study the computational complexity of planar valued constraint satisfaction problems (VCSPs). First, we show that intractable Boolean VCSPs have to be self-complementary to be tractable in the planar setting, thus extending a corresponding result of Dvorak and Kupec [ICALP'15] from CSPs to VCSPs. Second, we give a complete complexity classification of conservative planar VCSPs on arbitrary finite domains. As it turns out, in this case planarity does not lead to any new tractable cases, and thus our classification is a sharpening of the classification of conservative VCSPs by Kolmogorov and Zivny [JACM'13].

Peter Fulla and Stanislav Zivny. On Planar Valued CSPs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fulla_et_al:LIPIcs.MFCS.2016.39, author = {Fulla, Peter and Zivny, Stanislav}, title = {{On Planar Valued CSPs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.39}, URN = {urn:nbn:de:0030-drops-64537}, doi = {10.4230/LIPIcs.MFCS.2016.39}, annote = {Keywords: constraint satisfaction, valued constraint satisfaction, planarity, polymorphisms, multimorphisms} }

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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 give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.

Alberto Larrauri and Stanislav Živný. Solving Promise Equations over Monoids and Groups. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 146:1-146:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{larrauri_et_al:LIPIcs.ICALP.2024.146, author = {Larrauri, Alberto and \v{Z}ivn\'{y}, Stanislav}, title = {{Solving Promise Equations over Monoids and Groups}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {146:1--146:18}, 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.2024.146}, URN = {urn:nbn:de:0030-drops-202893}, doi = {10.4230/LIPIcs.ICALP.2024.146}, annote = {Keywords: constraint satisfaction, promise constraint satisfaction, equations, minions} }

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**Published in:** LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

General factors are a generalization of matchings. Given a graph G with a set π(v) of feasible degrees, called a degree constraint, for each vertex v of G, the general factor problem is to find a (spanning) subgraph F of G such that deg_F(v) ∈ π(v) for every v of G. When all degree constraints are symmetric Δ-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. Strongly polynomial-time algorithms are only known for weighted general factor problems that are reducible to the weighted matching problem by gadget constructions.
In this paper, we present a strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions. As an application, we obtain a strongly polynomial-time algorithm for the terminal backup problem by reducing it to the weighted general factor problem.

Shuai Shao and Stanislav Živný. A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{shao_et_al:LIPIcs.ISAAC.2023.57, author = {Shao, Shuai and \v{Z}ivn\'{y}, Stanislav}, title = {{A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees}}, booktitle = {34th International Symposium on Algorithms and Computation (ISAAC 2023)}, pages = {57:1--57:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-289-1}, ISSN = {1868-8969}, year = {2023}, volume = {283}, editor = {Iwata, Satoru and Kakimura, Naonori}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.57}, URN = {urn:nbn:de:0030-drops-193597}, doi = {10.4230/LIPIcs.ISAAC.2023.57}, annote = {Keywords: matchings, factors, edge constraint satisfaction problems, terminal backup problem, delta matroids} }

Document

**Published in:** Dagstuhl Reports, Volume 12, Issue 5 (2022)

Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes. CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena. For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non-trivially approximable, fixed-parameter tractable, or definable in a weak logic). In the last 15 years, research activity in this area has significantly intensified and hugely impressive progress was made. The Dagstuhl Seminar 22201 "The Constraint Satisfaction Problem: Complexity and Approximability" was aimed at bringing together researchers using all the different techniques in the study of the CSP so that they can share their insights obtained during the past four years. This report documents the material presented during the course of the seminar.

Martin Grohe, Venkatesan Guruswami, Dániel Marx, and Stanislav Živný. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201). In Dagstuhl Reports, Volume 12, Issue 5, pp. 112-130, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{grohe_et_al:DagRep.12.5.112, author = {Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav}, title = {{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 22201)}}, pages = {112--130}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {12}, number = {5}, editor = {Grohe, Martin and Guruswami, Venkatesan and Marx, D\'{a}niel and \v{Z}ivn\'{y}, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.5.112}, URN = {urn:nbn:de:0030-drops-174453}, doi = {10.4230/DagRep.12.5.112}, annote = {Keywords: Constraint satisfaction problem (CSP); Computational complexity; Hardness of approximation; Universal algebra; Semidefinite programming} }

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

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, …, k} to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results.
First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with k = O(√{nlog log n}/log n), where n is the number of vertices of the input hypergraph. This is established by building on ideas from algorithms designed for approximate graph colourings.
Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO 3-colouring for every constant uniformity r ≥ 5. In fact, we determine the precise relationship of polymorphism minions for all uniformities r ≥ 3, which reveals a key difference between r = 3,4 and r ≥ 5 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO (k+1)-colouring for LO k-colourable r-uniform hypergraphs for k ≥ 2 and r ≥ 5.

Tamio-Vesa Nakajima and Stanislav Živný. Linearly Ordered Colourings of Hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 128:1-128:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{nakajima_et_al:LIPIcs.ICALP.2022.128, author = {Nakajima, Tamio-Vesa and \v{Z}ivn\'{y}, Stanislav}, title = {{Linearly Ordered Colourings of Hypergraphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {128:1--128:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.128}, URN = {urn:nbn:de:0030-drops-164692}, doi = {10.4230/LIPIcs.ICALP.2022.128}, annote = {Keywords: hypegraph colourings, promise constraint satisfaction, PCSP, polymorphisms, minions, algebraic approach} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{larose_et_al:LIPIcs.ESA.2021.58, author = {Larose, Beno\^{i}t and Markovi\'{c}, Petar and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani and \v{Z}ivn\'{y}, Stanislav}, title = {{QCSP on Reflexive Tournaments}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {58:1--58:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.58}, URN = {urn:nbn:de:0030-drops-146392}, doi = {10.4230/LIPIcs.ESA.2021.58}, annote = {Keywords: computational complexity, algorithmic graph theory, quantified constraints, universal algebra, constraint satisfaction} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC'96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA'17] and non-Boolean domains by Butti and Živný [SIDMA'20].
Bansal, Svensson and Trevisan [FOCS'19] introduced a weaker notion of sparsification termed "additive sparsification", which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs.
As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P:{0,1}^k → {0,1} of a fixed arity k, we show that CSP(P)} admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P)} admits an additive sparsifier for any predicate P:D^k → {0,1} of a fixed arity k on an arbitrary finite domain D.

Eden Pelleg and Stanislav Živný. Additive Sparsification of CSPs. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{pelleg_et_al:LIPIcs.ESA.2021.75, author = {Pelleg, Eden and \v{Z}ivn\'{y}, Stanislav}, title = {{Additive Sparsification of CSPs}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {75:1--75:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.75}, URN = {urn:nbn:de:0030-drops-146562}, doi = {10.4230/LIPIcs.ESA.2021.75}, annote = {Keywords: additive sparsification, graphs, hypergraphs, minimum cuts} }

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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)

The promise constraint satisfaction problem (PCSP) is a recently introduced vast generalisation of the constraint satisfaction problem (CSP) that captures approximability of satisfiable instances. A PCSP instance comes with two forms of each constraint: a strict one and a weak one. Given the promise that a solution exists using the strict constraints, the task is to find a solution using the weak constraints. While there are by now several dichotomy results for fragments of PCSPs, they all consider (in some way) symmetric PCSPs.
1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric (non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami showed [SODA'18] that given a satisfiable instance of 1-in-3-SAT one can find a solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In other words, the PCSP template (𝟏-in-𝟑,NAE) is tractable.
We focus on non-symmetric PCSPs. In particular, we study PCSP templates obtained from the Boolean template (𝐭-in-𝐤, NAE) by either adding tuples to 𝐭-in-𝐤 or removing tuples from NAE. For the former, we classify all templates as either tractable or not solvable by the currently strongest known algorithm for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and Guruswami [SODA'20]. For the latter, we classify all templates as either tractable or NP-hard.

Alex Brandts and Stanislav Živný. Beyond PCSP(1-in-3, NAE). In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 121:1-121:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brandts_et_al:LIPIcs.ICALP.2021.121, author = {Brandts, Alex and \v{Z}ivn\'{y}, Stanislav}, title = {{Beyond PCSP(1-in-3, NAE)}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {121:1--121:14}, 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.2021.121}, URN = {urn:nbn:de:0030-drops-141902}, doi = {10.4230/LIPIcs.ICALP.2021.121}, annote = {Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).

Caterina Viola and Stanislav Živný. The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{viola_et_al:LIPIcs.MFCS.2020.85, author = {Viola, Caterina and \v{Z}ivn\'{y}, Stanislav}, title = {{The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {85:1--85:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.85}, URN = {urn:nbn:de:0030-drops-127566}, doi = {10.4230/LIPIcs.MFCS.2020.85}, annote = {Keywords: promise constraint satisfaction, valued constraint satisfaction, convex relaxations, polymorphisms} }

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Track A: Algorithms, Complexity and Games

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

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and Håstad [FOCS'14/SICOMP'17] proved a result known as "(2+ε)-SAT is NP-hard". They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains.

Alex Brandts, Marcin Wrochna, and Stanislav Živný. The Complexity of Promise SAT on Non-Boolean Domains. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 17:1-17:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brandts_et_al:LIPIcs.ICALP.2020.17, author = {Brandts, Alex and Wrochna, Marcin and \v{Z}ivn\'{y}, Stanislav}, title = {{The Complexity of Promise SAT on Non-Boolean Domains}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {17:1--17:13}, 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.2020.17}, URN = {urn:nbn:de:0030-drops-124241}, doi = {10.4230/LIPIcs.ICALP.2020.17}, annote = {Keywords: promise constraint satisfaction, PCSP, polymorphisms, algebraic approach, label cover} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

In this paper we study the complexity of counting Constraint Satisfaction Problems (CSPs) of the form #CSP(C,-), in which the goal is, given a relational structure A from a class C of structures and an arbitrary structure B, to find the number of homomorphisms from A to B. Flum and Grohe showed that #CSP(C,-) is solvable in polynomial time if C has bounded treewidth [FOCS'02]. Building on the work of Grohe [JACM'07] on decision CSPs, Dalmau and Jonsson then showed that, if C is a recursively enumerable class of relational structures of bounded arity, then assuming FPT != #W[1], there are no other cases of #CSP(C,-) solvable exactly in polynomial time (or even fixed-parameter time) [TCS'04].
We show that, assuming FPT != W[1] (under randomised parametrised reductions) and for C satisfying certain general conditions, #CSP(C,-) is not solvable even approximately for C of unbounded treewidth; that is, there is no fixed parameter tractable (and thus also not fully polynomial) randomised approximation scheme for #CSP(C,-). In particular, our condition generalises the case when C is closed under taking minors.

Andrei A. Bulatov and Stanislav Živný. Approximate Counting CSP Seen from the Other Side. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 60:1-60:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bulatov_et_al:LIPIcs.MFCS.2019.60, author = {Bulatov, Andrei A. and \v{Z}ivn\'{y}, Stanislav}, title = {{Approximate Counting CSP Seen from the Other Side}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {60:1--60:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.60}, URN = {urn:nbn:de:0030-drops-110041}, doi = {10.4230/LIPIcs.MFCS.2019.60}, annote = {Keywords: constraint satisfaction, approximate counting, homomorphisms} }

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

A cut epsilon-sparsifier of a weighted graph G is a re-weighted subgraph of G of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of epsilon. Since their introduction by Benczúr and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.

Silvia Butti and Stanislav Živný. Sparsification of Binary CSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 17:1-17:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{butti_et_al:LIPIcs.STACS.2019.17, author = {Butti, Silvia and \v{Z}ivn\'{y}, Stanislav}, title = {{Sparsification of Binary CSPs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {17:1--17:8}, 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.2019.17}, URN = {urn:nbn:de:0030-drops-102564}, doi = {10.4230/LIPIcs.STACS.2019.17}, annote = {Keywords: constraint satisfaction problems, minimum cuts, sparsification} }

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

Fulla, Uppman, and Živný [ACM ToCT'18] established a dichotomy theorem for Boolean surjective general-valued constraint satisfaction problems (VCSPs), i.e., VCSPs on two-element domains in which both labels have to be used in a solution. This result, in addition to identifying the complexity frontier, features the discovery of a new non-trivial tractable case (called EDS) that does not appear in the non-surjective setting.
In this work, we go beyond Boolean domains. As our main result, we introduce a generalisation of EDS to arbitrary finite domains called SEDS (similar to EDS) and establish a conditional complexity classification of SEDS VCSPs based on a reduction to smaller domains. This gives a complete classification of SEDS VCSPs on three-element domains. The basis of our tractability result is a natural generalisation of the Min-Cut problem, in which only solutions of certain size (given by a lower and upper bound) are permitted. We show that all near-optimal solutions to this problem can be enumerated in polynomial time, which might be of independent interest.

Gregor Matl and Stanislav Živný. Beyond Boolean Surjective VCSPs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{matl_et_al:LIPIcs.STACS.2019.52, author = {Matl, Gregor and \v{Z}ivn\'{y}, Stanislav}, title = {{Beyond Boolean Surjective VCSPs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {52:1--52:15}, 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.2019.52}, URN = {urn:nbn:de:0030-drops-102911}, doi = {10.4230/LIPIcs.STACS.2019.52}, annote = {Keywords: constraint satisfaction problems, valued constraint satisfaction, surjective constraint satisfaction, graph cuts} }

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**Published in:** Dagstuhl Reports, Volume 8, Issue 6 (2019)

Constraint satisfaction has always played a central role in computational
complexity theory; appropriate versions of CSPs are classical complete
problems for most standard complexity classes. CSPs constitute a very rich and
yet sufficiently manageable class of problems to give a good perspective on
general computational phenomena. For instance, they help to understand which
mathematical properties make a computational problem tractable (in a wide
sense, e.g., polynomial-time solvable, non-trivially approximable,
fixed-parameter tractable, or definable in a weak logic). In the last decade,
research activity in this area has significantly intensified and hugely
impressive progress was made.
The Dagstuhl Seminar 18231 "The Constraint Satisfaction Problem: Complexity and
Approximability" was aimed at bringing together researchers using all the
different techniques in the study of the CSP so that they can share their
insights obtained during the past three years. This report documents the
material presented during the course of the seminar.

Martin Grohe, Venkatesan Guruswami, and Stanislav Zivny. The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231). In Dagstuhl Reports, Volume 8, Issue 6, pp. 1-18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{grohe_et_al:DagRep.8.6.1, author = {Grohe, Martin and Guruswami, Venkatesan and Zivny, Stanislav}, title = {{The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)}}, pages = {1--18}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {8}, number = {6}, editor = {Grohe, Martin and Guruswami, Venkatesan and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.6.1}, URN = {urn:nbn:de:0030-drops-100251}, doi = {10.4230/DagRep.8.6.1}, annote = {Keywords: Constraint satisfaction problem (CSP); Computational complexity; CSP dichotomy conjecture; Hardness of approximation; Unique games conjecture; Parameterised complexity; Descriptive complexity; Universal algebra; Logic; Semidefinite programming} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Singleton arc consistency is an important type of local consistency which has been recently shown to solve all constraint satisfaction problems (CSPs) over constraint languages of bounded width. We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints. We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT. Combined with simple counter-examples for other patterns, we make significant progress towards a complete classification.

Clément Carbonnel, David A. Cohen, Martin C. Cooper, and Stanislav Zivny. On Singleton Arc Consistency for CSPs Defined by Monotone Patterns. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{carbonnel_et_al:LIPIcs.STACS.2018.19, author = {Carbonnel, Cl\'{e}ment and Cohen, David A. and Cooper, Martin C. and Zivny, Stanislav}, title = {{On Singleton Arc Consistency for CSPs Defined by Monotone Patterns}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.19}, URN = {urn:nbn:de:0030-drops-84876}, doi = {10.4230/LIPIcs.STACS.2018.19}, annote = {Keywords: constraint satisfaction problems, forbidden patterns, singleton arc consistency} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

A binary VCSP is a general framework for the minimization problem of a function represented as the sum of unary and binary cost functions.An important line of VCSP research is to investigate what functions can be solved in polynomial time.
Cooper-Zivny classified the tractability of binary VCSP instances according to the concept of "triangle,"
and showed that the only interesting tractable case is the one induced by the joint winner property (JWP).
Recently, Iwamasa-Murota-Zivny made a link between VCSP and discrete convex analysis, showing that a function satisfying the JWP can be transformed into a function represented as the sum of two M-convex functions, which can be minimized in polynomial time via an M-convex intersection algorithm if the value oracle of each M-convex function is given.
In this paper,
we give an algorithmic answer to a natural question: What binary finite-valued CSP instances can be solved in polynomial time via an M-convex intersection algorithm?
We solve this problem by devising a polynomial-time algorithm for obtaining a concrete form of the representation in the representable case.
Our result presents a larger tractable class of binary finite-valued CSPs, which properly contains the JWP class.

Hiroshi Hirai, Yuni Iwamasa, Kazuo Murota, and Stanislav Zivny. Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hirai_et_al:LIPIcs.STACS.2018.39, author = {Hirai, Hiroshi and Iwamasa, Yuni and Murota, Kazuo and Zivny, Stanislav}, title = {{Beyond JWP: A Tractable Class of Binary VCSPs via M-Convex Intersection}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.39}, URN = {urn:nbn:de:0030-drops-85042}, doi = {10.4230/LIPIcs.STACS.2018.39}, annote = {Keywords: valued constraint satisfaction problems, discrete convex analysis, M-convexity} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with the objective function given as a sum of fixed-arity functions; the values are rational numbers or infinity.
In Boolean surjective VCSPs variables take on labels from D={0,1} and an optimal assignment is required to use both labels from D. A classic example is the global min-cut problem in graphs. Building on the work of Uppman, we establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs. The newly discovered tractable case has an interesting structure related to projections of downsets and upsets. Our work generalises the dichotomy for {0,infinity}-valued constraint languages corresponding to CSPs) obtained by Creignou and Hebrard, and the dichotomy for {0,1}-valued constraint languages (corresponding to Min-CSPs) obtained by Uppman.

Peter Fulla and Stanislav Zivny. The Complexity of Boolean Surjective General-Valued CSPs. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fulla_et_al:LIPIcs.MFCS.2017.4, author = {Fulla, Peter and Zivny, Stanislav}, title = {{The Complexity of Boolean Surjective General-Valued CSPs}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.4}, URN = {urn:nbn:de:0030-drops-80623}, doi = {10.4230/LIPIcs.MFCS.2017.4}, annote = {Keywords: constraint satisfaction problems, surjective CSP, valued CSP, min-cut, polymorphisms, multimorphisms} }

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Complete Volume

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

DFU, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability, Complete Volume

The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Collection{DFU.Vol7.15301, title = {{DFU, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability, Complete Volume}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, 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}, URN = {urn:nbn:de:0030-drops-69752}, doi = {10.4230/DFU.Vol7.15301}, annote = {Keywords: Nonnumerical Algorithms and Problems} }

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

Front Matter, Table of Contents, Preface, List of Authors

The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{krokhin_et_al:DFU.Vol7.15301.i, author = {Krokhin, Andrei and Zivny, Stanislav}, title = {{Front Matter, Table of Contents, Preface, List of Authors}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {0:i--0:xii}, 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.i}, URN = {urn:nbn:de:0030-drops-69702}, doi = {10.4230/DFU.Vol7.15301.i}, annote = {Keywords: Front Matter, Table of Contents, Preface, List of Authors} }

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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 hybrid constraint satisfaction problems (CSPs) and valued constraint satisfaction problems (VCSPs). These are classes of (V)CSPs defined by restrictions that are not exclusively language-based or structure-based.

Martin C. Cooper and Stanislav Zivny. Hybrid Tractable Classes of Constraint Problems. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 113-135, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{cooper_et_al:DFU.Vol7.15301.113, author = {Cooper, Martin C. and Zivny, Stanislav}, title = {{Hybrid Tractable Classes of Constraint Problems}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {113--135}, 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.Vol7.15301.113}, URN = {urn:nbn:de:0030-drops-69616}, doi = {10.4230/DFU.Vol7.15301.113}, annote = {Keywords: Constraint satisfaction problems, Optimisation, Tractability} }

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

We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems (VCSPs). We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape.

Andrei Krokhin and Stanislav Zivny. The Complexity of Valued CSPs. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 233-266, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{krokhin_et_al:DFU.Vol7.15301.233, author = {Krokhin, Andrei and Zivny, Stanislav}, title = {{The Complexity of Valued CSPs}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {233--266}, 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.Vol7.15301.233}, URN = {urn:nbn:de:0030-drops-69665}, doi = {10.4230/DFU.Vol7.15301.233}, annote = {Keywords: Constraint satisfaction problems, Optimisation, Tractability} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We study the computational complexity of planar valued constraint satisfaction problems (VCSPs). First, we show that intractable Boolean VCSPs have to be self-complementary to be tractable in the planar setting, thus extending a corresponding result of Dvorak and Kupec [ICALP'15] from CSPs to VCSPs. Second, we give a complete complexity classification of conservative planar VCSPs on arbitrary finite domains. As it turns out, in this case planarity does not lead to any new tractable cases, and thus our classification is a sharpening of the classification of conservative VCSPs by Kolmogorov and Zivny [JACM'13].

Peter Fulla and Stanislav Zivny. On Planar Valued CSPs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 39:1-39:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fulla_et_al:LIPIcs.MFCS.2016.39, author = {Fulla, Peter and Zivny, Stanislav}, title = {{On Planar Valued CSPs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {39:1--39:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.39}, URN = {urn:nbn:de:0030-drops-64537}, doi = {10.4230/LIPIcs.MFCS.2016.39}, annote = {Keywords: constraint satisfaction, valued constraint satisfaction, planarity, polymorphisms, multimorphisms} }