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82
On the algebraic structure of combinatorial problems
 THEORETICAL COMPUTER SCIENCE
, 1998
"... ..."
Constraint Satisfaction Problems And Finite Algebras
, 1999
"... Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types c ..."
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Cited by 51 (7 self)
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Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. In this paper we show that any restricted set of constraint types can be associated with a finite universal algebra. We explore how the computational complexity of a restricted constraint satisfaction problem is connected to properties of the corresponding algebra. For this, we introduce a notion of `tractable algebra' and study how the tractability of an algebra relates to the tractability of its smaller derived algebras, including its subalgebras and homomorphic images. This allows us to significantly reduce the types of algebras which need to be investigated. Using these results we exhibit a common structural property of all known intractable constraint satisfaction problems. Finally, we classify all finite strictly simple surjective algebras wit...
Playing with Boolean blocks, part I: Post’s lattice with applications to complexity theory
 SIGACT News
"... Let us imagine children playing with a box containing a large number of building blocks such as LEGO TM, fischertechnik ® , Polydron, or something similar. Each block belongs to a certain class (given by, e. g., color, shape, or size) and usually the number of different such classes is relatively sm ..."
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Cited by 45 (13 self)
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Let us imagine children playing with a box containing a large number of building blocks such as LEGO TM, fischertechnik ® , Polydron, or something similar. Each block belongs to a certain class (given by, e. g., color, shape, or size) and usually the number of different such classes is relatively small. It is amazing to see how involved the constructions are that can be built by the kids. From
The Complexity Of Maximal Constraint Languages
, 2001
"... Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a constraint language and ..."
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Cited by 35 (8 self)
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Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a constraint language and the complexity of the problems it can express. In the present paper we systematically study the complexity of all maximal constraint languages, that is, languages whose expressive power is just weaker than that of the language of all constraints. Using the algebraic invariance properties of constraints, we exhibit a strong necessary condition for tractability of such a constraint language. Moreover, we show that, at least for small sets of values, this condition is also sufficient.
Shuffle on Trajectories: Syntactic Constraints
 Theor. Comp. Sci
, 1998
"... We introduce and investigate new methods to define parallel composition of words and languages. The operation of parallel composition leads to new shufflelike operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to ..."
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Cited by 25 (5 self)
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We introduce and investigate new methods to define parallel composition of words and languages. The operation of parallel composition leads to new shufflelike operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of noncontextfree languages in terms of contextfree and even regular languages. This paper concetrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution. TUCS Research Group
Constraints and Universal Algebra
 Annals of Mathematics and Artificial Intelligence
, 1998
"... In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two r ..."
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Cited by 20 (4 self)
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In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed...
On the Maximum Tolerable Noise for Reliable Computation by Formulas
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1995
"... It is shown that if a formula is constructed from noisy 2input NAND gates, with each gate failing independently with probability ", then reliable computation can or cannot take place according as "is less than or greater than" 0 = (3 \Gamma p 7)=4 = 0.08856... . ..."
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Cited by 19 (1 self)
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It is shown that if a formula is constructed from noisy 2input NAND gates, with each gate failing independently with probability ", then reliable computation can or cannot take place according as "is less than or greater than" 0 = (3 \Gamma p 7)=4 = 0.08856... .
The Complexity of Satisfiability Problems: Refining Schaefer’s Theorem
 J. Comput. Sys. Sci
"... problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and onl ..."
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Cited by 17 (7 self)
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problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.
The complexity of generalized satisfiability for linear temporal logic
 of Lecture Notes in Computer Science
, 2007
"... Abstract. In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NPcomplete or PSPACEcomplete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrea ..."
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Cited by 13 (9 self)
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Abstract. In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NPcomplete or PSPACEcomplete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post’s lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACEcomplete, NPcomplete, or in P. 2000 ACM Subject Classification:
Tractable Constraints Closed Under A Binary Operation
 Oxford University
, 2000
"... Many combinatorial search problems can be expressed as instances of the "constraint satisfaction problem" (CSP). This class of problems is known to be NPcomplete in general, so to ensure tractability it is natural to consider restricted subproblems in which the constraints have certain specified fo ..."
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Cited by 13 (7 self)
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Many combinatorial search problems can be expressed as instances of the "constraint satisfaction problem" (CSP). This class of problems is known to be NPcomplete in general, so to ensure tractability it is natural to consider restricted subproblems in which the constraints have certain specified forms. The algebraic approach to the CSP maintains that certain algebraic invariance properties of constraints can be used to determine the complexity of these restricted problems.