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Constraint Satisfaction Problems with Countable Homogeneous Templates
"... Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in ..."
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Cited by 24 (10 self)
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Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in temporal and spatial reasoning, infinitedimensional algebra, acyclic colorings in graph theory, artificial intelligence, phylogenetic reconstruction in computational biology, and tree descriptions in computational linguistics. We then give an introduction to the universalalgebraic approach to infinitedomain constraint satisfaction, and discuss how cores, polymorphism clones, and pseudovarieties can be used to study the computational complexity of CSPs with ωcategorical templates. The theoretical results will be illustrated by examples from the mentioned application areas. We close with a series of open problems and promising directions of future research.
The reducts of equality up to primitive positive interdefinability
 Journal of Symbolic Logic
"... Abstract. We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of ..."
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Abstract. We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain. Contents
Schaefer’s theorem for graphs
, 2011
"... Schaefer’s theorem is a complexity classification result for socalled Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NPcomplete. We present an analog of ..."
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Cited by 15 (12 self)
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Schaefer’s theorem is a complexity classification result for socalled Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NPcomplete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer’s result, the input consists of a set W of variables and a conjunction Φ of statements (“constraints”) about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifierfree firstorder formulas; the question is whether Φ is satisfiable in a graph. We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NPcomplete. This is achieved by a universalalgebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universalalgebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are firstorder definable in the countably infinite random graph. Our method to classify the computational complexity of those CSPs produces many statements of independent mathematical interest.
Decidability of definability
 IN: PROCEEDINGS OF THE 26TH ANNUAL IEEE SYMPOSIUM ON LOGIC IN COMPUTER SCIENCE (LICS’11), IEEE COMPUTER SOCIETY
, 2012
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On the scope of the universalalgebraic approach to constraint satisfaction
, 2009
"... The universalalgebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ωcategorical templates. Our first result is an exact charact ..."
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Cited by 12 (9 self)
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The universalalgebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ωcategorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an ωcategorical template. The universalalgebraic approach relies on the fact that in finite or ωcategorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. In this paper, we present results that can be used to study the computational complexity of CSPs with arbitrary infinite templates. Specifically, we prove that every CSP can be formulated with a template A such that a relation is primitive positive definable in A if and only if it is firstorder definable on A and preserved by the infinitary polymorphisms of A. We present applications of our general results to the description and analysis of the computational complexity of CSPs. In particular, we present a polymorphismbased description of those CSPs that are firstorder definable (and therefore can be solved in polynomialtime), and give general hardness criteria based on the absence of polymorphisms that depend on more than one argument.
ENUMERATING HOMOMORPHISMS
, 2009
"... 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 graphtheoretic ..."
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Cited by 8 (1 self)
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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 graphtheoretical 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.
All reducts of the random graph are modelcomplete
"... Abstract. We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or contains an operation that maps the random graph i ..."
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Cited by 6 (3 self)
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Abstract. We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or contains an operation that maps the random graph injectively to an induced subgraph which is a clique or an independent set. As a corollary, our result yields a new proof of Simon Thomas ’ classification of the five closed supergroups of the automorphism group of the random graph; our proof uses different Ramseytheoretic tools than the one given by Thomas, and is perhaps more straightforward. Since the monoids under consideration are endomorphism monoids of relational structures definable in the random graph, we are able to draw several modeltheoretic corollaries: One consequence of our result is that all structures with a firstorder definition in the random graph are modelcomplete. Moreover, we obtain a classification of these structures up to existential interdefinability. 1.
A FAST ALGORITHM AND DATALOG INEXPRESSIBILITY FOR TEMPORAL REASONING
, 2009
"... We introduce a new tractable temporal constraint language, which strictly contains the OrdHorn language of Bürkert and Nebel and the class of AND/OR precedence constraints. The algorithm we present for this language decides whether a given set of constraints is consistent in time that is quadratic ..."
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Cited by 6 (5 self)
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We introduce a new tractable temporal constraint language, which strictly contains the OrdHorn language of Bürkert and Nebel and the class of AND/OR precedence constraints. The algorithm we present for this language decides whether a given set of constraints is consistent in time that is quadratic in the input size. We also prove that (unlike OrdHorn) the constraint satisfaction problem of this language cannot be solved by Datalog or by establishing local consistency.
Relatively Quantified Constraint Satisfaction
"... The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more gener ..."
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Cited by 5 (2 self)
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The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more general framework of quantified constraint satisfaction, in which variables can be quantified both universally and existentially. We study the relatively quantified constraint satisfaction problem (RQCSP), in which the values for each individual variable can be arbitrarily restricted. We give a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.