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24
A survey of homogeneous structures
, 2010
"... A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as ..."
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A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as Fraissé limits of amalgamation classes of finite structures. The subject has connections to model theory, to permutation group theory, to combinatorics (for example through combinatorial enumeration, and through Ramsey theory), to descriptive set theory. Recently there has been a focus on connections to topological dynamics, and to constraint satisfaction. The article discusses connections between these topics, with an emphasis on examples, and on how special properties of an amalgamation class yield consequences for the automorphism group.
Reducts of Ramsey structures
"... One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the selfembedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Su ..."
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Cited by 23 (12 self)
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One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the selfembedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Such functions can be particularly well understood when the relational structure is countably in nite and has a rstorder de nition in another relational structure which has a nite language, is totally ordered and homogeneous, and has the Ramsey property. This is because in this situation, Ramsey theory provides the combinatorial tool for analyzing these functions { in a certain sense, it allows to represent such functions by functions on nite sets. This is a survey of results in model theory and theoretical computer science obtained recently by the authors in this context. In model theory, we approach the problem of classifying the reducts of countably in nite ordered homogeneous Ramsey structures in a nite language, and certain decidability
Dualities for constraint satisfaction problems
"... In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the nonexistence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfact ..."
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In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the nonexistence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfaction problems: finite duality, bounded pathwidth duality, and bounded treewidth duality.
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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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.
Qualitative temporal and spatial reasoning revisited
 In 16th EACSL Annual Conference on Computer Science and Logic (CSL’07
, 2007
"... Abstract. Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. In this area, one of the central questions for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a ..."
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Abstract. Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. In this area, one of the central questions for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a constraint language Γ has this “localtoglobal ” property implies polynomialtime tractability of the constraint language, and has further pleasant algorithmic consequences. In the present paper, we study the “localtoglobal ” property by making use of a recently established connection of this property with universal algebra. Specifically, the connection shows that this property is equivalent to the presence of a socalled quasi nearunanimity polymorphism of the constraint language. We obtain new algorithmic results and give very concise proofs of previously known theorems. Our results concern wellknown and heavily studied formalisms such as the point algebra and its extensions, Allen’s interval algebra, and the spatial reasoning language RCC5. 1
Nondichotomies in constraint satisfaction complexity
, 2008
"... We show that every computational decision problem is polynomialtime equivalent to a constraint satisfaction problem (CSP) with an infinite template. We also construct for every decision problem L an ωcategorical template Γ such that L reduces to CSP(Γ) and CSP(Γ) is in coNP L (i.e., the class coN ..."
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We show that every computational decision problem is polynomialtime equivalent to a constraint satisfaction problem (CSP) with an infinite template. We also construct for every decision problem L an ωcategorical template Γ such that L reduces to CSP(Γ) and CSP(Γ) is in coNP L (i.e., the class coNP with an oracle for L). CSPs with ωcategorical templates are of special interest, because the universalalgebraic approach can be applied to study their computational complexity. Furthermore, we prove that there are ωcategorical templates with coNPcomplete CSPs and ωcategorical templates with coNPintermediate CSPs, i.e., problems in coNP that are neither coNPcomplete nor in P (unless P=coNP). To construct the coNPintermediate CSP with ωcategorical template we modify the proof of Ladner’s theorem. A similar modification allows us to also prove a nondichotomy result for a class of lefthand side restricted CSPs, which was left open in [10]. We finally show that if the socalled localglobal conjecture for infinite constraint languages (over a finite domain) is false, then there is no dichotomy for the constraint satisfaction problem for infinite constraint languages.
A Note On Projective Graphs
, 2003
"... We show that all graphs with a simple extension property are projective. As a consequence of this result we settle in the armative a conjecture of Larose and Tardif and characterize all homogeneous graphs which are projective. ..."
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We show that all graphs with a simple extension property are projective. As a consequence of this result we settle in the armative a conjecture of Larose and Tardif and characterize all homogeneous graphs which are projective.
Topological Birkhoff
 Transactions of the American Mathematical Society
"... Abstract. One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. ..."
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Abstract. One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A, even if B is finite. We show that by considering the natural topology on the functions ofA and B in addition to the equations that hold between them, one can do with finite powers even for many interesting infinite algebras A. More precisely, we prove that if A and B are at most countable algebras which are oligomorphic, then the mapping which sends each function from A to the corresponding function in B preserves equations and is continuous if and only if B is a homomorphic image of a subalgebra of a finite power of A. Our result has the following consequences in model theory and in theoretical computer science: two ωcategorical structures are primitive positive biinterpretable if and only if their topological polymorphism clones are isomorphic. In particular, the complexity of the constraint satisfaction problem of an ωcategorical structure only depends on its topological polymorphism clone. 1.
The complexity of rooted phylogeny problems
 In Database Theory  ICDT 2010, 13th International Conference (2010
"... ABSTRACT Several computational problems in phylogenetic reconstruction can be formulated as restrictions of the following general problem: given a formula in conjunctive normal form where the atomic formulas are rooted triples, is there a rooted binary tree that satisfies the formula? If the formul ..."
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ABSTRACT Several computational problems in phylogenetic reconstruction can be formulated as restrictions of the following general problem: given a formula in conjunctive normal form where the atomic formulas are rooted triples, is there a rooted binary tree that satisfies the formula? If the formulas do not contain disjunctions and negations, the problem becomes the famous rooted triple consistency problem, which can be solved in polynomial time by an algorithm of Aho, Sagiv, Szymanski, and Ullman. If the clauses in the formulas are restricted to disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem remains NPcomplete. We systematically study the computational complexity of the problem for all such restrictions of the clauses in the input formula. For certain restricted disjunctions of triples we present an algorithm that has subquadratic running time and is asymptotically as fast as the fastest known algorithm for the rooted triple consistency problem. We also show that any restriction of the general rooted phylogeny problem that does not fall into our tractable class is NPcomplete, using known results about the complexity of Boolean constraint satisfaction problems. Finally, we present a pebble game argument that shows that the rooted triple consistency problem (and also all generalizations studied in this paper) cannot be solved by Datalog.
Compactness and its implications for qualitative spatial and temporal reasoning
 PROCEEDINGS OF THE 13TH INTERNATIONAL CONFERENCE ON PRINCIPLES OF KNOWLEDGE REPRESENTATION AND REASONING (KR
, 2012
"... A constraint satisfaction problem has compactness if any infinite set of constraints is satisfiable whenever all its finite subsets are satisfiable. We prove a sufficient condition for compactness, which holds for a range of problems including those based on the wellknown Interval Algebra (IA) and ..."
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A constraint satisfaction problem has compactness if any infinite set of constraints is satisfiable whenever all its finite subsets are satisfiable. We prove a sufficient condition for compactness, which holds for a range of problems including those based on the wellknown Interval Algebra (IA) and RCC8. Furthermore, we show that compactness leads to a useful necessary and sufficient condition for the recently introduced patchwork property, namely that patchwork holds exactly when every satisfiable finite network (i.e., set of constraints) has a canonical solution, that is, a solution that can be extended to a solution for any satisfiable finite extension of the network. Applying these general theorems to qualitative reasoning, we obtain important new results as well as significant strengthenings of previous results regarding IA, RCC8, and their fragments and extensions. In particular, we show that all the maximal tractable fragments of IA and RCC8 (containing the base relations) have patchwork and canonical solutions as long as networks are algebraically closed.