R. Dechter, Constraint networks, Encyclopedia of Artificial Intelligence (2nd Ed.), John Wiley, New York, 1991 pp. 276-285.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....reduces to finding an atomic reduction. x3. Comparison to classical constraint networks The notion of constraint network over a relation algebra includes as a special case the classical notion of network of binary relations. In the terminology and notation of Montanari [Mo74] see also [Mac87]) a network R of binary relations is a collection of sets X = fX 1 ; Xng, together with a binary relation R ij X i Theta X j for every pair of sets X i ; X j 2 X, such that R ii fhx; xi : x 2 X i g for i = 1; n. Such a network R represents the n ary relation R X 1 Theta ....

, Constraint satisfaction, Encyclopedia of Artificial Intelligence, ed. S. Shapiro, Wiley Interscience, 1987, pp. 205--211.

....zeroes to the left of it are removed. We provide a set of definitions and results relating the to the number of solutions (under the same partial assignment ) to CSPs composed out of the binary representations of the CPTs (see Figure 1) Basic definitions related to CSPs can be found in [Dechter, 1992] . Definition (Zero one layer of a CPT) The zero one layer of a CPT is a table of zeroes and ones derived from the Weight = 1 2 Weight = 1 4 Weight = 1 8 0.4 0 1 1 CPTs Values of Parents Node Values L A Sample Bayesian Network A CPT (Family) in the Bayes Net zero one layers precision ....

Dechter R. Constraint Networks. Encyclopedia of Artificial Intelligence, second edition, Wiley and Sons, Pages: 276-285, 1992.

....is to use heuristics derived from a maximum cardinality ordering (m ordering) Tarjan and Yannakakis, 1984] over the constraint network relating the variables of the system. Such techniques have been used in a variety of domains Bayesian network reasoning, constraint satisfaction problems [Dechter, 1992] etc. A constraint network on the variables of the system is defined by having the variables represent nodes and constraints in represent hyper edges. Any kind of optimization or satisfaction defined over the variables can be done in time exponential in the induced width of the graph [Dechter, ....

....1992] etc. A constraint network on the variables of the system is defined by having the variables represent nodes and constraints in represent hyper edges. Any kind of optimization or satisfaction defined over the variables can be done in time exponential in the induced width of the graph [Dechter, 1992] . Although the induced width itself cannot be found constructively in polynomial time, heuristics derived from m ordering perform reasonably well in practice. Throughout the rest of this paper, we will refer to all such heuristics as naive m ordering (naive because they do not supplement the ....

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Dechter R. Constraint Networks. Encyclopedia of Artificial Intelligence, second edition, Wiley and Sons, Pages: 276-285, 1992.

....rich user experience by integrating needs identification, product brokering, and product comparison poses challenges for the system architecture and interaction design. SMARTCLIENT ARCHITECTURE We have patented a technique that formulates travel planning as a constraint satisfaction problem (CSP [13,20]) It allows transferring product information between product server and buyers through a skinny data connection. At the buyer s side, information is assembled into product configurations according to their constraints and preferences. In travel planning, when a buyer contacts the flight server, ....

Mackworth, A., Constraint Satisfaction, Encyclopedia of Artificial Intelligence, pp. 205-211, John Wiley and Sons, 1987.

....as a way of describing certain combinatorial problems arising in image processing; it was quickly realized that the same general model was applicable to a much wider class of problems. The general problem has since been intensively studied, both theoretically and experimentally. The references [34], 35] 36] and [37] address general issues on this topic. Formally, a CSP P is specified by a tuple P = X, D, R 1 (S 1 ) R n (S n ) Synchronizing interactive web documents with FD Java constraints 11 where X is a finite set of variables D is a finite set of values (the domain ....

....of variables D is a finite set of values (the domain of P) Each pair R i (S i ) is a constraint. A solution for P is an assignment of values from D to each of the variables in X, which satisfies all the constraints simultaneously. Thus, problems modeled by means of constraint satisfaction [34] are represented by a finite set of variables, a set of finite domains associated with the variables, and a set of relations (or constraints) that limit the values that the variables can take. The main idea behind a CSP resolution is to assign only a value to each variable if the value is ....

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K. Macworth. Constraint satisfaction. S. C. Shapiro (ed.), Encyclopedia of Artificial Intelligence, vol. 1, pp. 285-293, Wiley Interscience, 1992.

....viewed as a measure of the device connectivity. This means that compiling devices for the purpose of answering diagnostic queries can be achieved under the same guarantees that one finds in graph based algorithms, which have been investigated extensively in the probabilistic [22, 20] constraint [16] and graph theoretic [1] literature. 2 Compiling a Device Consider the simple device shown in Figure 3 which has one input A and one output C. The circles enclosing A and C declare these variables as being observables; that is, device variables about which we plan to collect observations. Given ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

.... problem paradigm are computationally intractable (NP hard) Over the last two decades, a great deal of theoretical and experimental research has been focused on developing algorithms for solving constraint satisfaction problems and on identifying restricted subclasses that are tractable [27,86,125]. Techniques for processing constraints can be classified roughly as inference or search, and these approaches interact. Inference methods (such as the path and arc consistency techniques described below) enforce various forms of local consistency that add inferred problem constraints, which can ....

A. K. Mackworth. Constraint satisfaction. Encyclopedia of Artificial Intelligence, pages 285--293, 1992.

.... problem paradigm are computationally intractable (NP hard) Over the last two decades, a great deal of theoretical and experimental research has been focused on developing algorithms for solving constraint satisfaction problems and on identifying restricted subclasses that are tractable [27,86,125]. Techniques for processing constraints can be classified roughly as inference or search, and these approaches interact. Inference methods (such as the path and arc consistency techniques described below) enforce various forms of local consistency that add inferred problem constraints, which can ....

....understanding how to integrate them into a general CP framework are challenging research topics. Structure driven algorithms Problem structure can be characterized and exploited at the micro level (the structure of the constraints) and the macro level (the structure of the constraint network) [27,37]. Many structure driven techniques emerged from the topological characterization of tractable problems described in the next section. Various graph based techniques whose complexities are tied to graph parameters were identified. Even when the macro structure of the original problem does not have ....

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R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

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R. Dechter, Constraint networks, Encyclopedia of Artificial Intelligence (2nd Ed.), John Wiley, New York, 1991 pp. 276-285.

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R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

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R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276-- 285, 1992.

....networks. It is known that the dual graph of a constraint network transforms any non binary network into a binary one, where the domains of the variables are the allowed tuples in each relation and the constraints of the dual problem force equality over shared variables labeling the arcs [Dechter1992] Constraint propagation algorithms is a class of polynomial time algorithms that are at the center of constraint processing techniques. They were investigated extensively in the past three decades and the most well known versions are arc , path , and i consistency [Dechter1992] DEFINITION 2.2 ....

....labeling the arcs [Dechter1992] Constraint propagation algorithms is a class of polynomial time algorithms that are at the center of constraint processing techniques. They were investigated extensively in the past three decades and the most well known versions are arc , path , and i consistency [Dechter1992] DEFINITION 2.2 (arc consistency) Mackworth1977] Given a binary constraint network (X, D,C) the network is arc consistent iff for every binary constraint R ij C, every value v D i has a value u D j s.t. v, u) R ij . When a binary constraint network is not arc consistent, ....

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R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

....scheme S has topological properties that permit solution in polynomial time, N S is identifiable. We say that a scheme S is tractable when there exists a polynomial algorithm for deciding the consistency of every constraint network in the class N S . For instance, any tree or acyclic hypergraph [10] is a tractable scheme. 12 Theorem 3.13 Let S be a tractable scheme. Then the class of networks N S is strongly identifiable. Proof: The projection 5 S (ae) provides a tightest N S approximation to ae and can be computed in polynomial time. The tractability of S assures that the equality jrel(5 ....

R. Dechter, Constraint networks, Encyclopedia of Artificial Intelligence, (2nd ed.) (Wiley, New York, 1992) 276-285.

.... green,red green,red green,red Figure 1: A graph coloring example nation algorithms. 2. 1 Bucket elimination for constraints Constraint networks have been shown to be useful in formulating diverse problems such as scene labeling, scheduling, natural language parsing and temporal reasoning [13]. Consider the following graph coloring problem in Figure 1. The task is to assign a color to each node in the graph so that adjacent nodes will have different colors. Here is one way to solve this problem. Consider node E first. It can be colored either green or red. Since only two colors are ....

.... minimal induced width plus one [2] As noted before, the established connection between buckets sizes and induced width motivates finding an ordering with a smallest induced width, a task known to be hard [2] However, useful greedy heuristics as well as approximation algorithms are available [13, 4, 49]. In summary, the complexity of algorithm elim bel is dominated by the time and space needed to process a bucket. Recording a function on all the bucket s variables is time and space exponential in the number of variables mentioned in the bucket. The induced width bounds the arity of the ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

.... propagation outperforms integer programming (see for instance [26, 8] In relation to constraint processing algorithms, the Mini Bucket heuristics can be viewed as an extension of bounded constraint propagation algorithms that were investigated in the constraint community in the last decade [2]. Rather than applying this idea to the constraints only, we extended it here to the objective function as well. We mentioned earlier related work on specific MPE algorithms. It would be interesting to compare our algorithms with search based algorithms for MPE. Search is normally not the method ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276-- 285, 1992.

.... propagation outperforms integer programming (see for instance [25, 9] 40 In relation to constraint processing algorithms, the Mini Bucket heuristics can be viewed as an extension of bounded constraint propagation algorithms that were investigated in the constraint community in the last decade [3]. Rather than applying this idea to the constraints only, we extended it here to the objective function as well. We mentioned earlier related work on specific MPE algorithms. It would be interesting to compare our algorithms with search based algorithms for MPE. Search is normally not the method ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276-- 285, 1992.

.... and use relaxation to linear programming, assuming the integer restrictions on the domains are removed [ 19] The Mini Bucket heuristics can also be viewed as an extension of bounded constraint propagation algorithms that were investigated in the constraint community in the last decade [1]. However, rather than applying this idea to the constraints only, we extend it to the objective function as well. 2 Background 2.1 Notation and Definitions Constraint Satisfaction is a framework for formulating real world problems as a set of constraints between variables. They are graphically ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276-- 285, 1992.

....In section 8 we highlight more recent research which focus on combining the different improvement ideas into hybrid algorithms and present such hybrids. The final section provides historical remarks. Previous surveys on constraint processing as well as on backtracking algorithms can be found in [Dec92, Mac92, Kum92, Tsa93, KvB97]. 2 Definitions A constraint network R is a set of n variables X = fx 1 ; xng, a set of value domains D i for each variable x i , and a set of constraints or relations. Each value domain is a finite set of values, one of which must be assigned to the corresponding variable. A constraint ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

....relationships, emphasizing their syntactic description as bucket elimination algorithm. 2. 1 Bucket elimination for constraints Constraint networks have been shown to be useful in formulating such diverse problems as scene labeling, scheduling, natural language parsing and temporal reasoning [13]. Consider the following graph coloring problem in Figure 1. The task is to assign a color to each node in the graph so that adjacent nodes will have different colors. Here is one way to solve this problem. Consider node E first. It can be colored either green or red. Since only two colors are ....

....[2] As noted before, the established connection between buckets sizes and induced width motivates finding an ordering with a smallest induced width. Finding an ordering with the smallest induced width is hard [2] but useful greedy heuristics as well as approximation algorithms are available [13, 4]. In summary, the complexity of algorithm elim bel is dominated by the time and space needed to process a bucket. Recording a function on all the bucket s variables is time and space exponential in the number of variables mentioned in the bucket. The induced width bounds the arity of the ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

....nodes, X 1 ; D 1 ; X 2 ) and (D 1 ; X 1 ; X 2 ) where X t denotes an arbitrary sequence of x t i , and D t denotes action variable at time t. Note that the induced width of the first ordering is 3, while for the second ordering it is 2. An ordering heuristic, called min width[8], would prefer the second ordering which corresponds to shifting maximization operation in front of L d operators in our example. For similar example with N decision epochs, the complexity of dynamic programming on traditional ordering (X 1 ; D 1 ; X N01 ; D N01 ; X N ) i.e. ....

R. Dechter, Constraint networks, Encyclopedia of Artificial Intelligence (2nd Ed.), John Wiley, New York, 1991 pp. 276-285.

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R. Dechter, Constraint networks. Encyclopedia of Artificial Intelligence (2nd Ed.), John Wiley, New York, 1991 pp. 276-285.

....of the graph s cycles. A typical cycle cutset method enumerates the possible assignments to a set of cutset variables and, for each cutset assignment, solves (or reasons about) a tree like problem in polynomial time. Thus, the overall time complexity is exponential in the size of the cycle cutset [3]. Fortunately, enumerating all the cutset s assignments can be accomplished in linear space, yielding an overall linear space algorithm. The first question is which method, tree clustering or the cycle cutset scheme provide a better worst case time guarantees. This question was answered by Bertele ....

.... the moral graph s induced width (plus 1) In Figure 1b, the maximal cliques of the chordal graph are f (A,B) B,C,D) B,D,G) G,D,E,F) H,G,F,E)g, resulting in the join tree structure given in Figure 3(a) For more information on structuring a join tree, induced width and jointree clustering see [11,12,3,13]. A tighter bound on the space complexity of tree clustering may be obtained using the separator width. The separator width of a join tree is the maximal size of the intersections between any two cliques, and the separator width of a graph is the minimal separator width over all the graph s ....

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R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

.... approach applies the paradigm that heuristics can be generated by consulting relaxed models, suggested in [Pearl, 1984] The mini bucket heuristics can also be viewed as an extension of bounded constraint propagation algorithms that were investigated in the constraint community in the last decade [Dechter, 1992]. Here is some related work for finding the most probable explanation in Bayesian networks. It is known that solving the MPE task is NP hard. Complete algorithms for MPE use either the cycle cutset (also called conditioning) technique or the join tree clustering A B C D A B C D (a) b) E E ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

.... Nevertheless, it is this restrictiveness that allowed the developments of very useful concepts such as constraint propagation (also know as consistency enforcement ) through which various tractable subclasses had emerged and by which general purpose algorithms such as backtracking were improved [8,6,7,2]. However, real life problems frequently call for extending the basic model to allow nondeterminism as the representation of preferences among solutions. Such extensions relate the CSP model to known models for combinatorial optimization developed in the Operation Research community as well as to ....

R. Dechter. Constraint networks. Encyclopedia of Artificial Intelligence, pages 276--285, 1992.

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Dechter, R., Constraint Networks, Encyclopedia of Artificial Intelligence, Second Edition, 276-285, (1992).

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