Closure properties of constraints

Jeavons, Peter; Cohen, David A.; Gyssens, Marc

doi:10.1145/263867.263489

Cited by 424 publications

(391 citation statements)

References 21 publications

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“…In [2] (see [7] for a simpler proof), it was shown that for every Malt'sev operation ϕ, CSP(Inv(ϕ)) is solvable in polynomial time. This general result encompasses some previously known tractable cases of the CSP, such as CSP with constraints defined by a system of linear equations [23] or CSP with near-subgroups and its cosets [19,18].…”

Section: Dalmaumentioning

confidence: 70%

Generalized Majority-Minority Operations are Tractable

Dalmau¹

2006

Logical Methods in Computer Science

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Abstract. Generalized majority-minority (GMM) operations are introduced as a common generalization of near unanimity operations and Mal'tsev operations on finite sets. We show that every instance of the constraint satisfaction problem (CSP), where all constraint relations are invariant under a (fixed) GMM operation, is solvable in polynomial time. This constitutes one of the largest tractable cases of the CSP.

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Section: Dalmaumentioning

confidence: 70%

Generalized Majority-Minority Operations are Tractable

Dalmau¹

2006

Logical Methods in Computer Science

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“…This work has led to the identification of a number of classes of constraints which are tractable, in the sense that there exists a polynomial time algorithm to determine whether or not any collection of constraints from such a class can be simultaneously satisfied [2,12,20,27]. One powerful result in this area establishes that any tractable class of constraints over a finite domain must be preserved by a non-trivial algebraic operation, known as a polymorphism [4,19,20].…”

Section: Examplementioning

confidence: 99%

“…For example, the notion of a multimorphism can be used to characterise tractable subproblems in all of the following areas: in the case of the Satisfiability problem these include the Horn-Sat and 2-Sat subproblems [15]; in the case of the standard constraint satisfaction problem these include generalisations of Horn-Sat (such as the so-called 'max-closed' constraints [23,20]), generalisations of 2-Sat (such as the so-called '0/1/all' or 'implicative' constraints [8,18,25]) and systems of linear equations [20]; in the case of the optimisation problem Max-Sat these include the '0-valid' and '2-monotone' constraints [9]; in the case of optimisation problems over sets these include the submodular set functions [17,26] and bisubmodular set functions [14].…”

Section: Examplementioning

confidence: 99%

“…For crisp constraints, it has been show that the expressive power of a set of relations is determined by certain algebraic invariance properties of those relations, known as polymorphisms [4,20,22,21,28]. The concept of a polymorphism is specific to relations, and cannot be applied directly to the functions in a valued constraint language.…”

Section: Proposition 1 Let γ and γ Be Valued Constraint Languagesmentioning

confidence: 99%

“…If this is true, then multimorphisms are likely to play a central role in the analysis of complexity for soft constraints, just as the related notion of a polymorphism does in the analysis of complexity for crisp constraints [3][4][5][20][21][22] …”

Section: Majority/minority Multimorphismsmentioning

confidence: 99%

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Soft Constraints: Complexity and Multimorphisms

Cohen

Cooper

Jeavons

et al. 2003

Principles and Practice of Constraint Programming – CP 2003

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Abstract. Over the past few years there has been considerable progress in methods to systematically analyse the complexity of classical (crisp) constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of classical constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we begin a systematic investigation of the complexity of combinatorial optimisation problems expressed using various forms of soft constraints. We extend the notion of a polymorphism by introducing a more general algebraic operation, which we call a multimorphism. We show that a number of maximal tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms.

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Constraint Programming

Mathieu¹,

Keisu²

2000

Logic‐Based Methods for Optimization

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Constraint programming is a powerful paradigm for solving combinatorial search problems that draws on a wide range of techniques from artificial intelligence, operations research, algorithms, graph theory and elsewhere. The basic idea in constraint programming is that the user states the constraints and a general purpose constraint solver is used to solve them. Constraints are just relations, and a constraint satisfaction problem (CSP) states which relations should hold among the given decision variables. More formally, a constraint satisfaction problem consists of a set of variables, each with some domain of values, and a set of relations on subsets of these variables. For example, in scheduling exams at an university, the decision variables might be the times and locations of the different exams, and the constraints might be on the capacity of each examination room (e.g., we cannot schedule more students to sit exams in a given room at any one time than the room's capacity) and on the exams scheduled at the same time (e.g., we cannot schedule two exams at the same time if they share students in common). Constraint solvers take a real-world problem like this represented in terms of decision variables and constraints, and find an assignment to all the variables that satisfies the constraints. Extensions of this framework may involve, for example, finding optimal solutions according to one or more optimization criterion (e.g., minimizing the number of days over which exams need to be scheduled), finding all solutions, replacing (some or all) constraints with preferences, and considering a distributed setting where constraints are distributed among several agents.Constraint solvers search the solution space systematically, as with backtracking or branch and bound algorithms, or use forms of local search which may be incomplete. Systematic method often interleave search (see Section 4.3) and inference, where inference consists of propagating the information contained in one constraint to the neighboring constraints (see Section 4.2). Such inference reduces the parts of the search space that need to be visited. Special propagation procedures can be devised to suit specific constraints (called global constraints), which occur often in real life. Such global constraints are an important component in the success of constraint pro- Backtracking SearchA backtracking search for a solution to a CSP can be seen as performing a depth-first traversal of a search tree. This search tree is generated as the search progresses. At a

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Closure properties of constraints

Cited by 424 publications

References 21 publications

Generalized Majority-Minority Operations are Tractable

Generalized Majority-Minority Operations are Tractable

Soft Constraints: Complexity and Multimorphisms

Constraint Programming

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