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16
Universal algebra and hardness results for constraint satisfaction problems
, 2007
"... Abstract. We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give nonexpressibility results for various restrictions of Datalog. Furthermore, we show tha ..."
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Abstract. We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ) for complexity classes L, NL, P, NP and ModpL. These criteria also give nonexpressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not firstorder definable then it is Lhard. Our proofs rely on tame congruence theory and on a finegrain analysis of the complexity of reductions used in the algebraic study of CSP. The results pave the way for a refinement of the dichotomy conjecture stating that each CSP(Γ) lies in P or is NPcomplete and they match the recent classification of [2] for Boolean CSP. We also infer a partial classification theorem for the complexity of CSP(Γ) when the associated algebra of Γ is the full idempotent reduct of a preprimal algebra. Constraint satisfaction problems (CSP) provide a unifying framework to study various computational problems arising naturally in artificial intelligence, combinatorial optimization, graph homomorphisms and database theory. An in
Datalog and constraint satisfaction with infinite templates
 In Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS’06), LNCS 3884
, 2006
"... Abstract. On finite structures, there is a wellknown connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infin ..."
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Cited by 38 (20 self)
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Abstract. On finite structures, there is a wellknown connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates, i.e., for all computational problems that are closed under disjoint unions and whose complement is closed under homomorphisms. If the template Γ is ωcategorical, we obtain alternative characterizations of bounded Datalog width. We also show that CSP(Γ) can be solved in polynomial time if Γ is ωcategorical and the input is restricted to instances of bounded treewidth. Finally, we prove algebraic characterisations of those ωcategorical templates whose CSP has Datalog width (1, k), and for those whose CSP has strict Datalog width k.
Linear datalog and bounded path duality of relational structures
 Logical Methods in Computer Science
"... Abstract. In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathemat ..."
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Abstract. In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem CSP(B) with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL. 1.
Homomorphism Preservation Theorems
, 2008
"... The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in fin ..."
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Cited by 27 (0 self)
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The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the ̷Lo´sTarski theorem and Lyndon’s positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existentialpositive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a firstorder formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existentialpositive formula of equal quantifierrank.
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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Cited by 23 (8 self)
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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.
Weaker Forms of Monotonicity for Declarative Networking: a More Finegrained Answer to the CALMconjecture
"... The CALMconjecture, first stated by Hellerstein [23] and proved in its revised form by Ameloot et al. [13] within the framework of relational transducer networks, asserts that a query has a coordinationfree execution strategy if and only if the query is monotone. Zinn et al. [32] extended the fram ..."
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The CALMconjecture, first stated by Hellerstein [23] and proved in its revised form by Ameloot et al. [13] within the framework of relational transducer networks, asserts that a query has a coordinationfree execution strategy if and only if the query is monotone. Zinn et al. [32] extended the framework of relational transducer networks to allow for specific data distribution strategies and showed that the nonmonotone winmove query is coordinationfree for domainguided data distributions. In this paper, we complete the story by equating increasingly larger classes of coordinationfree computations with increasingly weaker forms of monotonicity and make Datalog variants explicit that capture each of these classes. One such fragment is based on stratified Datalog where rules are required to be connected with the exception of the last stratum. In addition, we characterize coordinationfreeness as those computations that do not require knowledge about all other nodes in the network, and therefore, can not globally coordinate. The results in this paper can be interpreted as a more finegrained answer to the CALMconjecture.
On Datalog vs. LFP
"... Abstract. We show that the homomorphism preservation theorem fails for LFP, both in general and in restriction to finite structures. That is, there is a formula of LFP that is preserved under homomorphisms (in the finite) but is not equivalent (in the finite) to a Datalog program. This resolves a qu ..."
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Abstract. We show that the homomorphism preservation theorem fails for LFP, both in general and in restriction to finite structures. That is, there is a formula of LFP that is preserved under homomorphisms (in the finite) but is not equivalent (in the finite) to a Datalog program. This resolves a question posed by Atserias. The results are established by two different methods: (1) a method of diagonalisation that works only in the presence of infinite structures, but establishes a stronger result showing a hierarchy of homomorphismpreserved problems in LFP; and (2) a method based on a pumping lemma for Datalog due to Afrati, Cosmadakis and Yannakakis which establishes the result in restriction to finite structures. We refine the pumping lemma of Afrati et al. and relate it to the power of Monadic SecondOrder Logic on tree decompositions of structures. 1
Constraint Satisfaction: A Personal Perspective
"... Attempts at classifying computational problems as polynomial time solvable, NPcomplete,or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes, ..."
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Attempts at classifying computational problems as polynomial time solvable, NPcomplete,or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes,
Descriptive complexity of approximate counting
"... Motivated by Fagin’s characterization of NP, Saluja et al. have introduced a logic based framework for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to nonnegative integers) is ’defined ’ by a firstorder sentence ϕ if for every instance A of ..."
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Motivated by Fagin’s characterization of NP, Saluja et al. have introduced a logic based framework for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to nonnegative integers) is ’defined ’ by a firstorder sentence ϕ if for every instance A of the problem, the number of possible satisfying assignments of the variables of ϕ in A is equal to C(A). The logic RHΠ1 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 #Phard, 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 RHΠ1. We prove that #CSP(H) is definable in monotone RHΠ1 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 RHΠ1, with the addition that H has, additionally, ⊤ (the greatest element of the lattice) as a polymorphism.