Results 1  10
of
84
Closure Properties of Constraints
 Journal of the ACM
, 1997
"... Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NPcomplete in general. In this paper we investigate the subclasses which arise from restricting the possible constraint types. We first show that any set of constrain ..."
Abstract

Cited by 133 (16 self)
 Add to MetaCart
Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NPcomplete in general. In this paper we investigate the subclasses which arise from restricting the possible constraint types. We first show that any set of constraints which does not give rise to an NPcomplete class of problems must satisfy a certain type of algebraic closure condition. We then investigate all the different possible forms of this algebraic closure property, and establish which of these are sufficient to ensure tractability. As examples, we show that all known classes of tractable constraints over finite domains can be characterised by such an algebraic closure property. Finally, we describe a simple computational procedure which can be used to determine the closure properties of a given set of constraints. This procedure involves solving a particular constraint satisfaction problem, which we call an `indicator problem'. Keywords: Cons...
Possibility theory in constraint satisfaction problems: Handling priority, preference and uncertainty
 Applied Intelligence
, 1996
"... In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preferenc ..."
Abstract

Cited by 74 (14 self)
 Add to MetaCart
In classical Constraint Satisfaction Problems (CSPs) knowledge is embedded in a set of hard constraints, each one restricting the possible values of a set of variables. However constraints in real world problems are seldom hard, and CSP's are often idealizations that do not account for the preference among feasible solutions. Moreover some constraints may have priority over others. Lastly, constraints may involve uncertain parameters. This paper advocates the use of fuzzy sets and possibility theory as a realistic approach for the representation of these three aspects. Fuzzy constraints encompass both preference relations among possible instanciations and priorities among constraints. In a Fuzzy Constraint Satisfaction Problem (FCSP), a constraint is satisfied to a degree (rather than satisfied or not satisfied) and the acceptability of a potential solution becomes a gradual notion. Even if the FCSP is partially inconsistent, best instanciations are provided owing to the relaxation of ...
Local and global relational consistency
 THEORETICAL COMPUTER SCIENCE
, 1997
"... Local consistency has proven to be an important concept in the theory and practice of constraint networks. In this paper, we present a new definition of local consistency, called relational consistency. The new definition is relationbased, in contrast with the previous definition of local consiste ..."
Abstract

Cited by 60 (13 self)
 Add to MetaCart
Local consistency has proven to be an important concept in the theory and practice of constraint networks. In this paper, we present a new definition of local consistency, called relational consistency. The new definition is relationbased, in contrast with the previous definition of local consistency, which we characterize as variablebased. We show the conceptual power of the new definition by showing how it unifies known elimination operators such as resolution in theorem proving, joins in relational databases, and variable elimination for solving linear inequalities. Algorithms for enforcing various levels of relational consistency are introduced and analyzed. We also show the usefulness of the new definition in characterizing relationships between properties of constraint networks and the level of local consistency needed to ensure global consistency.
Characterising Tractable Constraints
 Artificial Intelligence
, 1994
"... Finding solutions to a binary constraint satisfaction problem is known to be an NPcomplete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. This paper considers restricted sets of contraints which are closed under pe ..."
Abstract

Cited by 57 (18 self)
 Add to MetaCart
Finding solutions to a binary constraint satisfaction problem is known to be an NPcomplete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. This paper considers restricted sets of contraints which are closed under permutation of the labels. We identify a set of constraints which gives rise to a class of tractable problems and give polynomial time algorithms for solving such problems, and for finding the equivalent minimal network. We also prove that the class of problems generated by any set of constraints not contained in this restricted set is NPcomplete. 1 Introduction Finding solutions to a constraint satisfaction problem is known to be an NPcomplete problem in general [11] even when the constraints are restricted to binary constraints. However, many of the problems which arise in practice have special properties which allow them to be solved efficiently. The question of identifying restrictions t...
Consistency Techniques for Continuous Constraints
 Constraints
, 1996
"... We consider constraint satisfaction problemswith variables in continuous,numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constrai ..."
Abstract

Cited by 56 (7 self)
 Add to MetaCart
We consider constraint satisfaction problemswith variables in continuous,numerical domains. Contrary to most existing techniques, which focus on computing one single optimal solution, we address the problem of computing a compact representation of the space of all solutions admitted by the constraints. In particular, we show how globally consistent (also called decomposable) labelings of a constraint satisfaction problem can be computed.
Duality and polynomial testing of tree homomorphisms
 Trans. Amer. Math. Soc
, 1996
"... Abstract. Let H be a fixed digraph. We consider the Hcolouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain treelike obstructions. Specifically, we say that H has tree duality if, ..."
Abstract

Cited by 53 (16 self)
 Add to MetaCart
Abstract. Let H be a fixed digraph. We consider the Hcolouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain treelike obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. Weprovethatif Hhas tree duality then the Hcolouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the Xproperty studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the Hcolouring problem is NPcomplete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial Hcolouring problems. If P � = NP, then no oriented triad H with an NPcomplete Hcolouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NPcomplete Hcolouring problems given in the companion paper has tree duality. 1.
Tractable Disjunctions of Linear Constraints: Basic Results and Applications to Temporal Reasoning
 Theoretical Computer Science
, 1996
"... We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez an ..."
Abstract

Cited by 49 (2 self)
 Add to MetaCart
We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez and McAloon. We show that deciding consistency of a set of constraints in this class can be done in polynomial time. We also present a variable elimination algorithm which is similar to Fourier's algorithm for linear inequalities. Finally, we use these results to provide new temporal reasoning algorithms for the OrdHorn subclass of Allen's interval formalism. We also show that there is no low level of local consistency that can guarantee global consistency for the OrdHorn subclass. This property distinguishes the OrdHorn subclass from the pointizable subclass (for which strong 5consistency is sufficient to guarantee global consistency), and the continuous endpoint subclass (for whi...
Tractable Constraints on Ordered Domains
 Artificial Intelligence
, 1995
"... Finding solutions to a constraint satisfaction problem is known to be an NPcomplete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a clas ..."
Abstract

Cited by 47 (15 self)
 Add to MetaCart
Finding solutions to a constraint satisfaction problem is known to be an NPcomplete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a class of tractable problems. This class generalizes the notion of a Horn formula in propositional logic to larger domain sizes. We give a polynomial time algorithm for solving such problems, and prove that the class of problems generated by any larger set of constraints is NPcomplete. 1 Introduction Combinatorial problems abound in Artificial Intelligence. Examples include planning, temporal reasoning, linedrawing labelling and circuit design. The Constraint Satisfaction Problem (CSP) [14] is a generic combinatorial problem which is widely studied in the AI community because it allows all of these problems to be expressed in a natural and direct way. Reduction operations [12, 10] and intellig...
On the Minimality and Global Consistency of RowConvex Constraint Networks
, 1992
"... Constraint networks have beenshown to be useful in formulating such diverse problems as scene labeling, natural language parsing, and temporal reasoning. Given a constraint network, we often wish to (i) nd a solution that satis es the constraints and (ii) nd the corresponding minimal network where t ..."
Abstract

Cited by 46 (3 self)
 Add to MetaCart
Constraint networks have beenshown to be useful in formulating such diverse problems as scene labeling, natural language parsing, and temporal reasoning. Given a constraint network, we often wish to (i) nd a solution that satis es the constraints and (ii) nd the corresponding minimal network where the constraints are as explicit as possible. Both tasks are known to be NPcomplete in the general case. Task (i) is usually solved using a backtracking algorithm, and task (ii) is often solved only approximately by enforcing various levels of local consistency. In this paper, we identify a property of binary constraints called row convexity and show its usefulness in deciding when a form of local consistency called path consistency is sufficient to guarantee that a network is both minimal and globally consistent. Globally consistent networks have the property that a solution can be found without backtracking. We show that one can test for the row convexity property e ciently and we show, by examining
Constraints, Consistency, and Closure
 Artificial Intelligence
, 1998
"... Although the constraint satisfaction problem is NPcomplete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper, we describe a simple algebraic property which characterises all possi ..."
Abstract

Cited by 46 (12 self)
 Add to MetaCart
Although the constraint satisfaction problem is NPcomplete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper, we describe a simple algebraic property which characterises all possible constraint types for which strong kconsistency is sufficient to ensure global consistency, for each k ? 2. We give a number of examples to illustrate the application of this result. 1 Introduction The constraint satisfaction problem provides a framework in which it is possible to express, in a natural way, many combinatorial problems encountered in artificial intelligence and elsewhere. The aim in a constraint satisfaction problem is to find an assignment of values to a given set of variables subject to constraints on the values which can be assigned simultaneously to certain specified subsets of variables. The constraint satisfaction problem is known to be an NPcomplete problem in ge...