The Complexity of Satisfiability Problems: Refining Schaefer's Theorem (original) (raw)
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The Complexity of Boolean Constraint Isomorphism
Lecture Notes in Computer Science, 2004
We consider the Boolean constraint isomorphism problem, that is, the problem of determining whether two sets of Boolean constraint applications can be made equivalent by renaming the variables. We show that depending on the set of allowed constraints, the problem is either coNP-hard and GI-hard, equivalent to graph isomorphism, or polynomial-time solvable. This establishes a complete classification of the complexity of the problem, and moreover, it identifies exactly all those cases in which Boolean constraint isomorphism is polynomial-time manyone equivalent to graph isomorphism, the best-known and best-examined isomorphism problem in theoretical computer science.
Non-dichotomies in Constraint Satisfaction Complexity
Automata, Languages and Programming, 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 ω-categorical 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 ω-categorical templates are of special interest, because the universal-algebraic approach can be applied to study their computational complexity. Furthermore, we prove that there are ω-categorical templates with coNP-complete CSPs and ω-categorical templates with coNP-intermediate CSPs, i.e., problems in coNP that are neither coNP-complete nor in P (unless P=coNP). To construct the coNP-intermediate CSP with ω-categorical template we modify the proof of Ladner's theorem. A similar modification allows us to also prove a non-dichotomy result for a class of left-hand side restricted CSPs, which was left open in . We finally show that if the so-called local-global 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.
J. Multiple Valued Log. Soft Comput., 2022
Constraint satisfaction problems (CSPs) are combinatorial problems with strong ties to universal algebra and clone theory. The recently proved CSP dichotomy theorem states that each finite-domain CSP is either solvable in polynomial time, or that it is NP-complete. However, among the intractable CSPs there is a seemingly large variance in how fast they can be solved by exponential-time algorithms, which cannot be explained by the classical algebraic approach based on polymorphisms. In this contribution we will survey an alternative approach based on partial polymorphisms, which is useful for studying the fine-grained complexity of NP-complete CSPs. Moreover, we will state and discuss some challenging open problems in this research field. 1 Algebraic Background We begin by providing a self-contained introduction to the underlying algebraic approach. The reader familiar with universal algebra and clone theory can safely skim the two following subsections. miguel.couceiro@{loria,Inria}...
Universal algebra and hardness results for constraint satisfaction problems
Theoretical Computer Science, 2009
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 non-expressibility results for various restrictions of Datalog. Furthermore, we show that if CSP(Γ) is not first-order definable then it is L-hard. Our proofs rely on tame congruence theory and on a fine-grain 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 NP-complete 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.
Solution-Graphs of Boolean Formulas and Isomorphism1
Journal on Satisfiability, Boolean Modeling and Computation, 2019
The solution-graph of a Boolean formula on n variables is the subgraph of the hypercube H n induced by the satisfying assignments of the formula. The structure of solution-graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures based on local search. Several authors have studied connectivity problems in such graphs focusing on how the structure of the original formula might affect the complexity of the connectivity problems in the solution-graph. In this paper we study the complexity of the isomorphism problem of solution-graphs of Boolean formulas. We consider the classes of formulas that arise in the CSP-setting and investigate how the complexity of the isomorphism problem depends on the formula type. We observe that for general formulas the solution-graph isomorphism problem can be solved in exponential time while in the cases of 2CNF formulas, as well as for CPSS formulas, the problem is in the counting complexity class C = P, a subclass of PSPACE. We also prove a strong property on the structure of solution-graphs of Horn formulas showing that they are just unions of partial cubes. In addition, we give a PSPACE lower bound for the problem on general Boolean functions. We prove that for 2CNF, as well as for CPSS formulas the solution-graph isomorphism problem is hard for C = P under polynomial time many-one reductions, thus matching the given upper bound.
The Complexity of Equality Constraint Languages
Theory of Computing Systems, 2008
We classify the computational complexity of all constraint satisfaction problems where the constraint language is preserved by all permutations of the domain. A constraint language is preserved by all permutations of the domain if and only if all the relations in the language can be defined by boolean combinations of the equality relation. We call the corresponding constraint languages equality constraint languages.
The complexity of soft constraint satisfaction
Artificial Intelligence, 2006
Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints. We launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation, which we call a multimorphism. We show that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms. We also show that a simple set of NP-hard constraints has very restricted multimorphisms. Finally, we use the notion of multimorphism to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.
The Complexity of Satisfiability Problems: P = NP = PSPACE
By definition, an X3SAT is satisfiable iff there exists a satisfying assignment to. This paper shows that is satisfiable iff it is reducible to a satisfying assignment. Namely, → =1 () such that ∧ =1 () is true for some ∈ { , }, and that =1 () = 0 ∧ 1 ∧ • • • ∧ˆ, in which is a consistent subset of =1 () and 0 , 1 ,. .. ,ˆare properly disjoint. That is, they have no literal index in common, e.g., = (9 ∧ 2) and = (5 ∧ 7 ∧ 1) are properly disjoint, while and = (9 ∧ 2 ∧ 1) are disjoint but not properly. In this manner, (0 ∧ 1 ∧ • • • ∧ˆ) becomes consistent and (∧ 0 ∧ 1 ∧ • • • ∧ˆ) true for some ∈ { , }. Therefore, is satisfiable. A quantified boolean formula over a 3SAT is true iff is reducible to a satisfying assignment subject to universal quantification. Reducibility is due to the syntactic definition of the satisfiability of a clause. This reducibility is briefly illustrated as follows. Consider a clause = (⊙ ⊙), denoting an exactly-1 disjunction ⊙ of literals. is satisfiable if exactly one of { , , } is true, which denotes the (conventional/ semantic) definition of its satisfiability. Also, is satisfiable if (∧ ∧) ∨ (∧ ∧) ∨ (∧ ∧) is reducible, denoting the syntactic definition. The syntactic definition leads to the collapse of via a literal, viz., ∧ (⊙ ⊙) ⊢ ∧ , underlying the reducibility of. As a result, satisfiability is tackled syntactically and without searching satisfying assignments. Deciding reducibility is easy. Therefore, P = NP = PSPACE.
SIAM Journal on Computing, 2008
The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints. This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: QUANTIFIED 2-SAT (Aspvall, Plass, and Tarjan 1979), QUANTIFIED HORN-SAT (Karpinski, Kleine Büning, and Schmitt 1987), and QUANTIFIED AFFINE-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size.
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
Logical Methods in Computer Science, 2012
The universal-algebraic 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) ω-categorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an ω-categorical template.