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**Published in:** LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

It is known that the weighted version of Edge Multiway Cut (also known as Multiterminal Cut) is NP-complete on planar graphs of maximum degree 3. In contrast, for the unweighted version, NP-completeness is only known for planar graphs of maximum degree 11. In fact, the complexity of unweighted Edge Multiway Cut was open for graphs of maximum degree 3 for over twenty years. We prove that the unweighted version is NP-complete even for planar graphs of maximum degree 3. As weighted Edge Multiway Cut is polynomial-time solvable for graphs of maximum degree at most 2, we have now closed the complexity gap. We also prove that (unweighted) Node Multiway Cut (both with and without deletable terminals) is NP-complete for planar graphs of maximum degree 3. By combining our results with known results, we can apply two meta-classifications on graph containment from the literature. This yields full dichotomies for all three problems on H-topological-minor-free graphs and, should H be finite, on H-subgraph-free graphs as well. Previously, such dichotomies were only implied for H-minor-free graphs.

Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{johnson_et_al:LIPIcs.SWAT.2024.29, author = {Johnson, Matthew and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Smith, Siani and van Leeuwen, Erik Jan}, title = {{Edge Multiway Cut and Node Multiway Cut Are Hard for Planar Subcubic Graphs}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {29:1--29:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.29}, URN = {urn:nbn:de:0030-drops-200699}, doi = {10.4230/LIPIcs.SWAT.2024.29}, annote = {Keywords: multiway cut, planar subcubic graph, complexity dichotomy, graph containment} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

For any finite set ℋ = {H_1,…,H_p} of graphs, a graph is ℋ-subgraph-free if it does not contain any of H_1,…,H_p as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of ℋ-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems.
We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

Matthew Johnson, Barnaby Martin, Sukanya Pandey, Daniël Paulusma, Siani Smith, and Erik Jan van Leeuwen. Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{johnson_et_al:LIPIcs.MFCS.2023.57, author = {Johnson, Matthew and Martin, Barnaby and Pandey, Sukanya and Paulusma, Dani\"{e}l and Smith, Siani and van Leeuwen, Erik Jan}, title = {{Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {57:1--57:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.57}, URN = {urn:nbn:de:0030-drops-185914}, doi = {10.4230/LIPIcs.MFCS.2023.57}, annote = {Keywords: forbidden subgraphs, independent feedback vertex set, treewidth} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Stabbing Planes is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system established via communication complexity arguments. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas.
In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner’s Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon’s combinatorial Nullenstellensatz.
We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Stefan Dantchev, Nicola Galesi, Abdul Ghani, and Barnaby Martin. Depth Lower Bounds in Stabbing Planes for Combinatorial Principles. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dantchev_et_al:LIPIcs.STACS.2022.24, author = {Dantchev, Stefan and Galesi, Nicola and Ghani, Abdul and Martin, Barnaby}, title = {{Depth Lower Bounds in Stabbing Planes for Combinatorial Principles}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {24:1--24:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.24}, URN = {urn:nbn:de:0030-drops-158349}, doi = {10.4230/LIPIcs.STACS.2022.24}, annote = {Keywords: proof complexity, computational complexity, lower bounds, cutting planes, stabbing planes} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

A natural way of increasing our understanding of NP-complete graph problems is to restrict the input to a special graph class. Classes of H-free graphs, that is, graphs that do not contain some graph H as an induced subgraph, have proven to be an ideal testbed for such a complexity study. However, if the forbidden graph H contains a cycle or claw, then these problems often stay NP-complete. A recent complexity study (MFCS 2019) on the k-Colouring problem shows that we may still obtain tractable results if we also bound the diameter of the H-free input graph. We continue this line of research by initiating a complexity study on the impact of bounding the diameter for a variety of classical vertex partitioning problems restricted to H-free graphs. We prove that bounding the diameter does not help for Independent Set, but leads to new tractable cases for problems closely related to 3-Colouring. That is, we show that Near-Bipartiteness, Independent Feedback Vertex Set, Independent Odd Cycle Transversal, Acyclic 3-Colouring and Star 3-Colouring are all polynomial-time solvable for chair-free graphs of bounded diameter. To obtain these results we exploit a new structural property of 3-colourable chair-free graphs.

Christoph Brause, Petr Golovach, Barnaby Martin, Daniël Paulusma, and Siani Smith. Partitioning H-Free Graphs of Bounded Diameter. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{brause_et_al:LIPIcs.ISAAC.2021.21, author = {Brause, Christoph and Golovach, Petr and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Partitioning H-Free Graphs of Bounded Diameter}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {21:1--21:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.21}, URN = {urn:nbn:de:0030-drops-154543}, doi = {10.4230/LIPIcs.ISAAC.2021.21}, annote = {Keywords: vertex partitioning problem, H-free, diameter, complexity dichotomy} }

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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 173, 28th Annual European Symposium on Algorithms (ESA 2020)

A k-colouring c of a graph G is a mapping V(G) → {1,2,… k} such that c(u) ≠ c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling).
A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years.
We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results:
1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs.
2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

Jan Bok, Nikola Jedlic̆ková, Barnaby Martin, Daniël Paulusma, and Siani Smith. Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bok_et_al:LIPIcs.ESA.2020.22, author = {Bok, Jan and Jedlic̆kov\'{a}, Nikola and Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {22:1--22:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.22}, URN = {urn:nbn:de:0030-drops-128885}, doi = {10.4230/LIPIcs.ESA.2020.22}, annote = {Keywords: acyclic colouring, star colouring, injective colouring, H-free, dichotomy} }

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

The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring H-Free Graphs of Bounded Diameter. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{martin_et_al:LIPIcs.MFCS.2019.14, author = {Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani}, title = {{Colouring H-Free Graphs of Bounded Diameter}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {14:1--14:14}, 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.2019.14}, URN = {urn:nbn:de:0030-drops-109584}, doi = {10.4230/LIPIcs.MFCS.2019.14}, annote = {Keywords: vertex colouring, H-free graph, diameter} }

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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Res(s) is an extension of Resolution working on s-DNFs. We prove tight n^{Omega(k)} lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [Massimo Lauria et al., 2017] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [Nathan Segerlind et al., 2004], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding.
We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHP^m_n for m>n. Using the the same recursive approach we prove the new result that for any delta>0, Bin-PHP^m_n requires proofs of size 2^{n^{1-delta}} in Res(s) for s=o(log^{1/2}n). Our lower bound is almost optimal since for m >= 2^{sqrt{n log n}} there are quasipolynomial size proofs of Bin-PHP^m_n in Res(log n).
Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in Pi_2-form and with no finite model.

Stefan Dantchev, Nicola Galesi, and Barnaby Martin. Resolution and the Binary Encoding of Combinatorial Principles. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 6:1-6:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dantchev_et_al:LIPIcs.CCC.2019.6, author = {Dantchev, Stefan and Galesi, Nicola and Martin, Barnaby}, title = {{Resolution and the Binary Encoding of Combinatorial Principles}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {6:1--6:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.6}, URN = {urn:nbn:de:0030-drops-108287}, doi = {10.4230/LIPIcs.CCC.2019.6}, annote = {Keywords: Proof complexity, k-DNF resolution, binary encodings, Clique and Pigeonhole principle} }

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

We apply the algebraic approach for Constraint Satisfaction Problems (CSPs) with counting quantifiers, developed by Bulatov and Hedayaty, for the first time to obtain classifications for computational complexity. We develop the consistency approach for expanding polymorphisms to deduce that, if H has an expanding majority polymorphism, then the corresponding CSP with counting quantifiers is tractable. We elaborate some applications of our result, in particular deriving a complexity classification for partially reflexive graphs endowed with all unary relations. For each such structure, either the corresponding CSP with counting quantifiers is in P, or it is NP-hard.

Florent R. Madelaine and Barnaby Martin. Consistency for Counting Quantifiers. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{madelaine_et_al:LIPIcs.MFCS.2018.11, author = {Madelaine, Florent R. and Martin, Barnaby}, title = {{Consistency for Counting Quantifiers}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {11:1--11:13}, 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.2018.11}, URN = {urn:nbn:de:0030-drops-95931}, doi = {10.4230/LIPIcs.MFCS.2018.11}, annote = {Keywords: Quantified Constraints, Constraint Satisfaction, Logic in Computer Science, Universal Algebra, Computational Complexity} }

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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:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C_4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

Barnaby Martin, Daniël Paulusma, and Erik Jan van Leeuwen. Disconnected Cuts in Claw-free Graphs. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 61:1-61:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{martin_et_al:LIPIcs.ESA.2018.61, author = {Martin, Barnaby and Paulusma, Dani\"{e}l and van Leeuwen, Erik Jan}, title = {{Disconnected Cuts in Claw-free Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {61:1--61:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah 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.2018.61}, URN = {urn:nbn:de:0030-drops-95249}, doi = {10.4230/LIPIcs.ESA.2018.61}, annote = {Keywords: disconnected cut, surjective homomorphism, biclique cover, claw-freeness} }

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

The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs.
Chen [2014] proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL, otherwise it is NP-complete.
By combining this result with some known and new results we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.

Benoit Larose, Barnaby Martin, and Daniel Paulusma. Surjective H-Colouring over Reflexive Digraphs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{larose_et_al:LIPIcs.STACS.2018.49, author = {Larose, Benoit and Martin, Barnaby and Paulusma, Daniel}, title = {{Surjective H-Colouring over Reflexive Digraphs}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {49:1--49:14}, 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.2018.49}, URN = {urn:nbn:de:0030-drops-84882}, doi = {10.4230/LIPIcs.STACS.2018.49}, annote = {Keywords: Surjective H-Coloring, Computational Complexity, Algorithmic Graph Theory, Universal Algebra, Constraint Satisfaction} }

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

Let A be an idempotent algebra on a finite domain. We combine results of Chen, Zhuk and Carvalho et al. to argue that if A satisfies the polynomially generated powers property (PGP), then QCSP(Inv(A)) is in NP. We then use the result of Zhuk to prove a converse, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture.
We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen as a generalisation of the already-known Collapsibility. For three-element domain algebras A that are Switchable, we prove that for every finite subset Delta of Inv(A), Pol(Delta) is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility.
Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.

Catarina Carvalho, Barnaby Martin, and Dmitriy Zhuk. The Complexity of Quantified Constraints Using the Algebraic Formulation. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{carvalho_et_al:LIPIcs.MFCS.2017.27, author = {Carvalho, Catarina and Martin, Barnaby and Zhuk, Dmitriy}, title = {{The Complexity of Quantified Constraints Using the Algebraic Formulation}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {27:1--27:14}, 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.2017.27}, URN = {urn:nbn:de:0030-drops-80793}, doi = {10.4230/LIPIcs.MFCS.2017.27}, annote = {Keywords: Quantified Constraints, Computational Complexity, Universal Algebra, Constraint Satisfaction} }

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

I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions.

Barnaby Martin. Quantified Constraints in Twenty Seventeen. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 327-346, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{martin:DFU.Vol7.15301.327, author = {Martin, Barnaby}, title = {{Quantified Constraints in Twenty Seventeen}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {327--346}, 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.Vol7.15301.327}, URN = {urn:nbn:de:0030-drops-69699}, doi = {10.4230/DFU.Vol7.15301.327}, annote = {Keywords: Quantified constraints, Constraint satisfaction problems, Computational complexity, Parameterized complexity, Universal algebra} }

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

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

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

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@InProceedings{bodirsky_et_al:LIPIcs.ICALP.2016.119, author = {Bodirsky, Manuel and Martin, Barnaby and Pinsker, Michael and Pongr\'{a}cz, Andr\'{a}s}, title = {{Constraint Satisfaction Problems for Reducts of Homogeneous Graphs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {119:1--119:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.119}, URN = {urn:nbn:de:0030-drops-62543}, doi = {10.4230/LIPIcs.ICALP.2016.119}, annote = {Keywords: Constraint Satisfaction, Homogeneous Graphs, Computational Complexity, Universal Algebra, Ramsey Theory} }

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**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

We study the computational problem of checking whether a logical sentence is true in a finite abelian group. We prove that model checking first-order sentences on finite abelian groups is fixed-parameter tractable, when parameterized by the size of the sentence. We also prove that model checking monadic second-order sentences on finite abelian groups finitely presented by integer matrices is not fixed-parameter tractable (under standard assumptions in parameterized complexity).

Simone Bova and Barnaby Martin. First-Order Queries on Finite Abelian Groups. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 41-59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bova_et_al:LIPIcs.CSL.2015.41, author = {Bova, Simone and Martin, Barnaby}, title = {{First-Order Queries on Finite Abelian Groups}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {41--59}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.41}, URN = {urn:nbn:de:0030-drops-54060}, doi = {10.4230/LIPIcs.CSL.2015.41}, annote = {Keywords: Finite Abelian Groups, First-Order Logic, Monadic Second-Order Logic} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9441, The Constraint Satisfaction Problem: Complexity and Approximability (2010)

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over fixed, finite structures B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). Extending the algebraic methods of a previous paper, we derive a complete complexity classification for these problems as B ranges over structures of domain size 4. Specifically, each problem is either in Logspace, is NP-complete, is co-NP-complete or is Pspace-complete.

Barnaby Martin and Jos Martin. The complexity of positive first-order logic without equality II: The four-element case. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Seminar Proceedings, Volume 9441, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{martin_et_al:DagSemProc.09441.5, author = {Martin, Barnaby and Martin, Jos}, title = {{The complexity of positive first-order logic without equality II: The four-element case}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {1--12}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9441}, editor = {Andrei A. Bulatov and Martin Grohe and Phokion G. Kolaitis and Andrei Krokhin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09441.5}, URN = {urn:nbn:de:0030-drops-23670}, doi = {10.4230/DagSemProc.09441.5}, annote = {Keywords: Quantified constraints, Galois connection} }

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