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Documents authored by Dalmau, Victor

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Dalmau, Victor

Document
DOI: 10.4230/LIPIcs.ICDT.2024.8
When Do Homomorphism Counts Help in Query Algorithms?

Authors: Balder ten Cate, Victor Dalmau, Phokion G. Kolaitis, and Wei-Lin Wu

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring ℕ of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring 𝔹, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over 𝔹 by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over 𝔹 and left query algorithms over ℕ. In general, there are properties that admit a left query algorithm over ℕ but not over 𝔹. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over 𝔹 if and only if it admits a left query algorithm over ℕ. In other words and rather surprisingly, homomorphism counts over ℕ do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over 𝔹 and a right query algorithm over 𝔹.
Cite as

Balder ten Cate, Victor Dalmau, Phokion G. Kolaitis, and Wei-Lin Wu. When Do Homomorphism Counts Help in Query Algorithms?. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2024.8,
  author =	{ten Cate, Balder and Dalmau, Victor and Kolaitis, Phokion G. and Wu, Wei-Lin},
  title =	{{When Do Homomorphism Counts Help in Query Algorithms?}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.8},
  URN =		{urn:nbn:de:0030-drops-197905},
  doi =		{10.4230/LIPIcs.ICDT.2024.8},
  annote =	{Keywords: query algorithms, homomorphism, homomorphism counts, conjunctive query, constraint satisfaction}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2024.10
Right-Adjoints for Datalog Programs

Authors: Balder ten Cate, Víctor Dalmau, and Jakub Opršal

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
A Datalog program can be viewed as a syntactic specification of a mapping from database instances over some schema to database instances over another schema. We establish a large class of Datalog programs for which this mapping admits a (generalized) right-adjoint. We employ these results to obtain new insights into the existence of, and methods for constructing, homomorphism dualities within restricted classes of instances. From this, we derive new results regarding the existence of uniquely characterizing data examples for database queries in the presence of integrity constraints.
Cite as

Balder ten Cate, Víctor Dalmau, and Jakub Opršal. Right-Adjoints for Datalog Programs. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2024.10,
  author =	{ten Cate, Balder and Dalmau, V{\'\i}ctor and Opr\v{s}al, Jakub},
  title =	{{Right-Adjoints for Datalog Programs}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.10},
  URN =		{urn:nbn:de:0030-drops-197929},
  doi =		{10.4230/LIPIcs.ICDT.2024.10},
  annote =	{Keywords: Datalog, Adjoints, Homomorphism Dualities, Database Constraints, Conjunctive Queries, Data Examples}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.27
Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

Authors: Silvia Butti and Víctor Dalmau

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
Given a pair of graphs 𝐀 and 𝐁, the problems of deciding whether there exists either a homomorphism or an isomorphism from 𝐀 to 𝐁 have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where 𝐀 and 𝐁 are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.
Cite as

Silvia Butti and Víctor Dalmau. Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{butti_et_al:LIPIcs.MFCS.2021.27,
  author =	{Butti, Silvia and Dalmau, V{\'\i}ctor},
  title =	{{Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.27},
  URN =		{urn:nbn:de:0030-drops-144679},
  doi =		{10.4230/LIPIcs.MFCS.2021.27},
  annote =	{Keywords: Weisfeiler-Leman algorithm, Sherali-Adams hierarchy, Graph homomorphism, Constraint Satisfaction Problem}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2021.9
Conjunctive Queries: Unique Characterizations and Exact Learnability

Authors: Balder ten Cate and Victor Dalmau

Published in: LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

Abstract
We answer the question of which conjunctive queries are uniquely characterized by polynomially many positive and negative examples, and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts.
Cite as

Balder ten Cate and Victor Dalmau. Conjunctive Queries: Unique Characterizations and Exact Learnability. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 9:1-9:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2021.9,
  author =	{ten Cate, Balder and Dalmau, Victor},
  title =	{{Conjunctive Queries: Unique Characterizations and Exact Learnability}},
  booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
  pages =	{9:1--9:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-179-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{186},
  editor =	{Yi, Ke and Wei, Zhewei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.9},
  URN =		{urn:nbn:de:0030-drops-137172},
  doi =		{10.4230/LIPIcs.ICDT.2021.9},
  annote =	{Keywords: Conjunctive Queries, Homomorphisms, Frontiers, Unique Characterizations, Exact Learnability, Schema Mappings, Description Logic}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.20
The Complexity of the Distributed Constraint Satisfaction Problem

Authors: Silvia Butti and Victor Dalmau

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
We study the complexity of the Distributed Constraint Satisfaction Problem (DCSP) on a synchronous, anonymous network from a theoretical standpoint. In this setting, variables and constraints are controlled by agents which communicate with each other by sending messages through fixed communication channels. Our results endorse the well-known fact from classical CSPs that the complexity of fixed-template computational problems depends on the template’s invariance under certain operations. Specifically, we show that DCSP(Γ) is polynomial-time tractable if and only if Γ is invariant under symmetric polymorphisms of all arities. Otherwise, there are no algorithms that solve DCSP(Γ) in finite time. We also show that the same condition holds for the search variant of DCSP. Collaterally, our results unveil a feature of the processes' neighbourhood in a distributed network, its iterated degree, which plays a major role in the analysis. We explore this notion establishing a tight connection with the basic linear programming relaxation of a CSP.
Cite as

Silvia Butti and Victor Dalmau. The Complexity of the Distributed Constraint Satisfaction Problem. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{butti_et_al:LIPIcs.STACS.2021.20,
  author =	{Butti, Silvia and Dalmau, Victor},
  title =	{{The Complexity of the Distributed Constraint Satisfaction Problem}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus 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.2021.20},
  URN =		{urn:nbn:de:0030-drops-136654},
  doi =		{10.4230/LIPIcs.STACS.2021.20},
  annote =	{Keywords: Constraint Satisfaction Problems, Distributed Algorithms, Polymorphisms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.29
Dismantlability, Connectedness, and Mixing in Relational Structures

Authors: Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, and Benoît Larose

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, statistical physics, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, topological properties of the solution set such as connectedness is related to the hardness of CSPs over random structures. In approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions or the free energy of spin systems. Additionally, in the decision CSPs, structural properties of the relational structures involved - like, for example, dismantlability - and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that the structural property of dismantlability of the target graph, the connectedness of the set of homomorphisms, good mixing properties of the corresponding spin system, and the uniqueness of Gibbs measure are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.
Cite as

Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, and Benoît Larose. Dismantlability, Connectedness, and Mixing in Relational Structures. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{briceno_et_al:LIPIcs.ICALP.2019.29,
  author =	{Brice\~{n}o, Raimundo and Bulatov, Andrei A. and Dalmau, V{\'\i}ctor and Larose, Beno\^{i}t},
  title =	{{Dismantlability, Connectedness, and Mixing in Relational Structures}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.29},
  URN =		{urn:nbn:de:0030-drops-106059},
  doi =		{10.4230/LIPIcs.ICALP.2019.29},
  annote =	{Keywords: relational structure, constraint satisfaction problem, homomorphism, mixing properties, Gibbs measure}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2015.161
The Product Homomorphism Problem and Applications

Authors: Balder ten Cate and Victor Dalmau

Published in: LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

Abstract
The product homomorphism problem (PHP) takes as input a finite collection of structures A_1, ..., A_n and a structure B, and asks if there is a homomorphism from the direct product between A_1, A_2, ..., and A_n, to B. We pinpoint the computational complexity of this problem. Our motivation stems from the fact that PHP naturally arises in different areas of database theory. In particular, it is equivalent to the problem of determining whether a relation is definable by a conjunctive query, and the existence of a schema mapping that fits a given collection of positive and negative data examples. We apply our results to obtain complexity bounds for these problems.
Cite as

Balder ten Cate and Victor Dalmau. The Product Homomorphism Problem and Applications. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 161-176, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2015.161,
  author =	{ten Cate, Balder and Dalmau, Victor},
  title =	{{The Product Homomorphism Problem and Applications}},
  booktitle =	{18th International Conference on Database Theory (ICDT 2015)},
  pages =	{161--176},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-79-8},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{31},
  editor =	{Arenas, Marcelo and Ugarte, Mart{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.161},
  URN =		{urn:nbn:de:0030-drops-49832},
  doi =		{10.4230/LIPIcs.ICDT.2015.161},
  annote =	{Keywords: Homomorphisms, Direct Product, Data Examples, Definability, Conjunctive Queries, Schema Mappings}
}
Document
DOI: 10.4230/LIPIcs.CSL.2013.149
Descriptive complexity of approximate counting CSPs

Authors: Andrei Bulatov, Victor Dalmau, and Marc Thurley

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Abstract
Motivated by Fagin's characterization of NP, Saluja et al. have introduced a logic based frame- work for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to non-negative integers) is `defined’ by a first-order sentence phi if for every instance A of the problem, the number of possible satisfying assignments of the variables of phi in A is equal to C(A). The logic RHPI_1 has been introduced by Dyer et al. in their study of the counting complexity class #BIS. The interest in the class #BIS stems from the fact that, it is quite plausible that the problems in #BIS are not #P-hard, nor they admit a fully polynomial randomized approximation scheme. In the present paper we investigate which counting constraint satisfaction problems #CSP(H) are definable in the monotone fragment of RHPI_1. We prove that #CSP(H) is definable in monotone RHPI_1 whenever H is invariant under meet and join operations of a distributive lattice. We prove that the converse also holds if H contains the equality relation. We also prove similar results for counting CSPs expressible by linear Datalog. The results in this case are very similar to those for monotone RHPI1, with the addition that H has, additionally, \top (the greatest element of the lattice) as a polymorphism.
Cite as

Andrei Bulatov, Victor Dalmau, and Marc Thurley. Descriptive complexity of approximate counting CSPs. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 149-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{bulatov_et_al:LIPIcs.CSL.2013.149,
  author =	{Bulatov, Andrei and Dalmau, Victor and Thurley, Marc},
  title =	{{Descriptive complexity of approximate counting CSPs}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{149--164},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.149},
  URN =		{urn:nbn:de:0030-drops-41955},
  doi =		{10.4230/LIPIcs.CSL.2013.149},
  annote =	{Keywords: Constraint Satisfaction Problems, Approximate Counting, Descriptive Complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1838
Enumerating Homomorphisms

Authors: Andrei A. Bulatov, Victor Dalmau, Martin Grohe, and Daniel Marx

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
The homomorphism problem for relational structures is an abstract way of formulating constraint satisfaction problems (CSP) and various problems in database theory. The decision version of the homomorphism problem received a lot of attention in literature; in particular, the way the graph-theoretical structure of the variables and constraints influences the complexity of the problem is intensively studied. Here we study the problem of enumerating all the solutions with polynomial delay from a similar point of view. It turns out that the enumeration problem behaves very differently from the decision version. We give evidence that it is unlikely that a characterization result similar to the decision version can be obtained. Nevertheless, we show nontrivial cases where enumeration can be done with polynomial delay.
Cite as

Andrei A. Bulatov, Victor Dalmau, Martin Grohe, and Daniel Marx. Enumerating Homomorphisms. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 231-242, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bulatov_et_al:LIPIcs.STACS.2009.1838,
  author =	{Bulatov, Andrei A. and Dalmau, Victor and Grohe, Martin and Marx, Daniel},
  title =	{{Enumerating Homomorphisms}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{231--242},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1838},
  URN =		{urn:nbn:de:0030-drops-18385},
  doi =		{10.4230/LIPIcs.STACS.2009.1838},
  annote =	{Keywords: }
}

Dalmau, Víctor

Document
DOI: 10.4230/LIPIcs.ICDT.2024.8
When Do Homomorphism Counts Help in Query Algorithms?

Authors: Balder ten Cate, Victor Dalmau, Phokion G. Kolaitis, and Wei-Lin Wu

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
A query algorithm based on homomorphism counts is a procedure for determining whether a given instance satisfies a property by counting homomorphisms between the given instance and finitely many predetermined instances. In a left query algorithm, we count homomorphisms from the predetermined instances to the given instance, while in a right query algorithm we count homomorphisms from the given instance to the predetermined instances. Homomorphisms are usually counted over the semiring ℕ of non-negative integers; it is also meaningful, however, to count homomorphisms over the Boolean semiring 𝔹, in which case the homomorphism count indicates whether or not a homomorphism exists. We first characterize the properties that admit a left query algorithm over 𝔹 by showing that these are precisely the properties that are both first-order definable and closed under homomorphic equivalence. After this, we turn attention to a comparison between left query algorithms over 𝔹 and left query algorithms over ℕ. In general, there are properties that admit a left query algorithm over ℕ but not over 𝔹. The main result of this paper asserts that if a property is closed under homomorphic equivalence, then that property admits a left query algorithm over 𝔹 if and only if it admits a left query algorithm over ℕ. In other words and rather surprisingly, homomorphism counts over ℕ do not help as regards properties that are closed under homomorphic equivalence. Finally, we characterize the properties that admit both a left query algorithm over 𝔹 and a right query algorithm over 𝔹.
Cite as

Balder ten Cate, Victor Dalmau, Phokion G. Kolaitis, and Wei-Lin Wu. When Do Homomorphism Counts Help in Query Algorithms?. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2024.8,
  author =	{ten Cate, Balder and Dalmau, Victor and Kolaitis, Phokion G. and Wu, Wei-Lin},
  title =	{{When Do Homomorphism Counts Help in Query Algorithms?}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.8},
  URN =		{urn:nbn:de:0030-drops-197905},
  doi =		{10.4230/LIPIcs.ICDT.2024.8},
  annote =	{Keywords: query algorithms, homomorphism, homomorphism counts, conjunctive query, constraint satisfaction}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2024.10
Right-Adjoints for Datalog Programs

Authors: Balder ten Cate, Víctor Dalmau, and Jakub Opršal

Published in: LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Abstract
A Datalog program can be viewed as a syntactic specification of a mapping from database instances over some schema to database instances over another schema. We establish a large class of Datalog programs for which this mapping admits a (generalized) right-adjoint. We employ these results to obtain new insights into the existence of, and methods for constructing, homomorphism dualities within restricted classes of instances. From this, we derive new results regarding the existence of uniquely characterizing data examples for database queries in the presence of integrity constraints.
Cite as

Balder ten Cate, Víctor Dalmau, and Jakub Opršal. Right-Adjoints for Datalog Programs. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2024.10,
  author =	{ten Cate, Balder and Dalmau, V{\'\i}ctor and Opr\v{s}al, Jakub},
  title =	{{Right-Adjoints for Datalog Programs}},
  booktitle =	{27th International Conference on Database Theory (ICDT 2024)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-312-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{290},
  editor =	{Cormode, Graham and Shekelyan, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.10},
  URN =		{urn:nbn:de:0030-drops-197929},
  doi =		{10.4230/LIPIcs.ICDT.2024.10},
  annote =	{Keywords: Datalog, Adjoints, Homomorphism Dualities, Database Constraints, Conjunctive Queries, Data Examples}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2021.27
Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem

Authors: Silvia Butti and Víctor Dalmau

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract
Given a pair of graphs 𝐀 and 𝐁, the problems of deciding whether there exists either a homomorphism or an isomorphism from 𝐀 to 𝐁 have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where 𝐀 and 𝐁 are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.
Cite as

Silvia Butti and Víctor Dalmau. Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{butti_et_al:LIPIcs.MFCS.2021.27,
  author =	{Butti, Silvia and Dalmau, V{\'\i}ctor},
  title =	{{Fractional Homomorphism, Weisfeiler-Leman Invariance, and the Sherali-Adams Hierarchy for the Constraint Satisfaction Problem}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.27},
  URN =		{urn:nbn:de:0030-drops-144679},
  doi =		{10.4230/LIPIcs.MFCS.2021.27},
  annote =	{Keywords: Weisfeiler-Leman algorithm, Sherali-Adams hierarchy, Graph homomorphism, Constraint Satisfaction Problem}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2021.9
Conjunctive Queries: Unique Characterizations and Exact Learnability

Authors: Balder ten Cate and Victor Dalmau

Published in: LIPIcs, Volume 186, 24th International Conference on Database Theory (ICDT 2021)

Abstract
We answer the question of which conjunctive queries are uniquely characterized by polynomially many positive and negative examples, and how to construct such examples efficiently. As a consequence, we obtain a new efficient exact learning algorithm for a class of conjunctive queries. At the core of our contributions lie two new polynomial-time algorithms for constructing frontiers in the homomorphism lattice of finite structures. We also discuss implications for the unique characterizability and learnability of schema mappings and of description logic concepts.
Cite as

Balder ten Cate and Victor Dalmau. Conjunctive Queries: Unique Characterizations and Exact Learnability. In 24th International Conference on Database Theory (ICDT 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 186, pp. 9:1-9:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2021.9,
  author =	{ten Cate, Balder and Dalmau, Victor},
  title =	{{Conjunctive Queries: Unique Characterizations and Exact Learnability}},
  booktitle =	{24th International Conference on Database Theory (ICDT 2021)},
  pages =	{9:1--9:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-179-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{186},
  editor =	{Yi, Ke and Wei, Zhewei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2021.9},
  URN =		{urn:nbn:de:0030-drops-137172},
  doi =		{10.4230/LIPIcs.ICDT.2021.9},
  annote =	{Keywords: Conjunctive Queries, Homomorphisms, Frontiers, Unique Characterizations, Exact Learnability, Schema Mappings, Description Logic}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.20
The Complexity of the Distributed Constraint Satisfaction Problem

Authors: Silvia Butti and Victor Dalmau

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
We study the complexity of the Distributed Constraint Satisfaction Problem (DCSP) on a synchronous, anonymous network from a theoretical standpoint. In this setting, variables and constraints are controlled by agents which communicate with each other by sending messages through fixed communication channels. Our results endorse the well-known fact from classical CSPs that the complexity of fixed-template computational problems depends on the template’s invariance under certain operations. Specifically, we show that DCSP(Γ) is polynomial-time tractable if and only if Γ is invariant under symmetric polymorphisms of all arities. Otherwise, there are no algorithms that solve DCSP(Γ) in finite time. We also show that the same condition holds for the search variant of DCSP. Collaterally, our results unveil a feature of the processes' neighbourhood in a distributed network, its iterated degree, which plays a major role in the analysis. We explore this notion establishing a tight connection with the basic linear programming relaxation of a CSP.
Cite as

Silvia Butti and Victor Dalmau. The Complexity of the Distributed Constraint Satisfaction Problem. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{butti_et_al:LIPIcs.STACS.2021.20,
  author =	{Butti, Silvia and Dalmau, Victor},
  title =	{{The Complexity of the Distributed Constraint Satisfaction Problem}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus 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.2021.20},
  URN =		{urn:nbn:de:0030-drops-136654},
  doi =		{10.4230/LIPIcs.STACS.2021.20},
  annote =	{Keywords: Constraint Satisfaction Problems, Distributed Algorithms, Polymorphisms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.29
Dismantlability, Connectedness, and Mixing in Relational Structures

Authors: Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, and Benoît Larose

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, statistical physics, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, topological properties of the solution set such as connectedness is related to the hardness of CSPs over random structures. In approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions or the free energy of spin systems. Additionally, in the decision CSPs, structural properties of the relational structures involved - like, for example, dismantlability - and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that the structural property of dismantlability of the target graph, the connectedness of the set of homomorphisms, good mixing properties of the corresponding spin system, and the uniqueness of Gibbs measure are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Briceño. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.
Cite as

Raimundo Briceño, Andrei A. Bulatov, Víctor Dalmau, and Benoît Larose. Dismantlability, Connectedness, and Mixing in Relational Structures. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{briceno_et_al:LIPIcs.ICALP.2019.29,
  author =	{Brice\~{n}o, Raimundo and Bulatov, Andrei A. and Dalmau, V{\'\i}ctor and Larose, Beno\^{i}t},
  title =	{{Dismantlability, Connectedness, and Mixing in Relational Structures}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{29:1--29:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.29},
  URN =		{urn:nbn:de:0030-drops-106059},
  doi =		{10.4230/LIPIcs.ICALP.2019.29},
  annote =	{Keywords: relational structure, constraint satisfaction problem, homomorphism, mixing properties, Gibbs measure}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2015.161
The Product Homomorphism Problem and Applications

Authors: Balder ten Cate and Victor Dalmau

Published in: LIPIcs, Volume 31, 18th International Conference on Database Theory (ICDT 2015)

Abstract
The product homomorphism problem (PHP) takes as input a finite collection of structures A_1, ..., A_n and a structure B, and asks if there is a homomorphism from the direct product between A_1, A_2, ..., and A_n, to B. We pinpoint the computational complexity of this problem. Our motivation stems from the fact that PHP naturally arises in different areas of database theory. In particular, it is equivalent to the problem of determining whether a relation is definable by a conjunctive query, and the existence of a schema mapping that fits a given collection of positive and negative data examples. We apply our results to obtain complexity bounds for these problems.
Cite as

Balder ten Cate and Victor Dalmau. The Product Homomorphism Problem and Applications. In 18th International Conference on Database Theory (ICDT 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 31, pp. 161-176, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{tencate_et_al:LIPIcs.ICDT.2015.161,
  author =	{ten Cate, Balder and Dalmau, Victor},
  title =	{{The Product Homomorphism Problem and Applications}},
  booktitle =	{18th International Conference on Database Theory (ICDT 2015)},
  pages =	{161--176},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-79-8},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{31},
  editor =	{Arenas, Marcelo and Ugarte, Mart{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2015.161},
  URN =		{urn:nbn:de:0030-drops-49832},
  doi =		{10.4230/LIPIcs.ICDT.2015.161},
  annote =	{Keywords: Homomorphisms, Direct Product, Data Examples, Definability, Conjunctive Queries, Schema Mappings}
}
Document
DOI: 10.4230/LIPIcs.CSL.2013.149
Descriptive complexity of approximate counting CSPs

Authors: Andrei Bulatov, Victor Dalmau, and Marc Thurley

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Abstract
Motivated by Fagin's characterization of NP, Saluja et al. have introduced a logic based frame- work for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to non-negative integers) is `defined’ by a first-order sentence phi if for every instance A of the problem, the number of possible satisfying assignments of the variables of phi in A is equal to C(A). The logic RHPI_1 has been introduced by Dyer et al. in their study of the counting complexity class #BIS. The interest in the class #BIS stems from the fact that, it is quite plausible that the problems in #BIS are not #P-hard, nor they admit a fully polynomial randomized approximation scheme. In the present paper we investigate which counting constraint satisfaction problems #CSP(H) are definable in the monotone fragment of RHPI_1. We prove that #CSP(H) is definable in monotone RHPI_1 whenever H is invariant under meet and join operations of a distributive lattice. We prove that the converse also holds if H contains the equality relation. We also prove similar results for counting CSPs expressible by linear Datalog. The results in this case are very similar to those for monotone RHPI1, with the addition that H has, additionally, \top (the greatest element of the lattice) as a polymorphism.
Cite as

Andrei Bulatov, Victor Dalmau, and Marc Thurley. Descriptive complexity of approximate counting CSPs. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 149-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{bulatov_et_al:LIPIcs.CSL.2013.149,
  author =	{Bulatov, Andrei and Dalmau, Victor and Thurley, Marc},
  title =	{{Descriptive complexity of approximate counting CSPs}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{149--164},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.149},
  URN =		{urn:nbn:de:0030-drops-41955},
  doi =		{10.4230/LIPIcs.CSL.2013.149},
  annote =	{Keywords: Constraint Satisfaction Problems, Approximate Counting, Descriptive Complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2009.1838
Enumerating Homomorphisms

Authors: Andrei A. Bulatov, Victor Dalmau, Martin Grohe, and Daniel Marx

Published in: LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Abstract
The homomorphism problem for relational structures is an abstract way of formulating constraint satisfaction problems (CSP) and various problems in database theory. The decision version of the homomorphism problem received a lot of attention in literature; in particular, the way the graph-theoretical structure of the variables and constraints influences the complexity of the problem is intensively studied. Here we study the problem of enumerating all the solutions with polynomial delay from a similar point of view. It turns out that the enumeration problem behaves very differently from the decision version. We give evidence that it is unlikely that a characterization result similar to the decision version can be obtained. Nevertheless, we show nontrivial cases where enumeration can be done with polynomial delay.
Cite as

Andrei A. Bulatov, Victor Dalmau, Martin Grohe, and Daniel Marx. Enumerating Homomorphisms. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 231-242, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{bulatov_et_al:LIPIcs.STACS.2009.1838,
  author =	{Bulatov, Andrei A. and Dalmau, Victor and Grohe, Martin and Marx, Daniel},
  title =	{{Enumerating Homomorphisms}},
  booktitle =	{26th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{231--242},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-09-5},
  ISSN =	{1868-8969},
  year =	{2009},
  volume =	{3},
  editor =	{Albers, Susanne and Marion, Jean-Yves},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1838},
  URN =		{urn:nbn:de:0030-drops-18385},
  doi =		{10.4230/LIPIcs.STACS.2009.1838},
  annote =	{Keywords: }
}
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