van Beek P. and Dechter R. On the minimality and global consistency of row convex constraint networks. In Journal of the ACM, 1995.  Home/Search   Document Details and Download   Summary   ACM   TOC   Related Articles   Check

....shown that when the constraints themselves are convex and binary [137, 38] path consistency ensures global consistency irrespective of the topology of the network. 2.5. CONSISTENCY TECHNIQUES 27 These simplifying properties deal with the type of constraints (monotone, functional, convex, etc. [99, 36, 38, 28, 137, 139, 141, 41]) topological characteristics of the constraint graph [47, 48] or a combination of both notions. Chapter 5 presents some results falling under the class of works identifying such properties. It contains a more extensive review of the subject. The complexity of consistency algorithms may also be ....

....no significant practical impact. Indeed, and analogously to the case of discrete row convex problems, enforcing 5 consistency may create intermediate constraints of arity four , which disables the application of Helly s Theorem to ternary networks. To overcome this limitation, van Beek and Dechter [139] have introduced the notion of relational path consistency for discrete problems: Definition 5.5 (van Beek Dechter) Let R be a network of relations over a set of variables X, and let R S and R T be two relations in R, where S; T X. We say that R S and R T are relationally path consistent ....

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van Beek P. and Dechter R. On the minimality and global consistency of row convex constraint networks. In Journal of the ACM, 1995.

....which is missing in the network. If we achieve arc consistency, some inconsistencies can be missed, that would be detected if C was handled as a single global constraint. Related work Relational arc consistency is another local consistency which has been proposed for non binary constraints [vBD95]. Unfortunately, it is very expensive when ECAI 98 Workshop Programme. used on practical problems with non binary constraints of great arity since relational arc consistency on a constraint C implies global consistency on the network restricted to the variables of C. Therefore, enforcing ....

P. van Beek and R. Dechter. On the minimality and global consistency of row-convex constraint networks. Journal of the ACM, 42(3):543561, 1995. ECAI98 Workshop on Non Binary Constraints 59

....minimal domains by enforcing full arc consistency on the reduced network. The complexity of all these tasks is dominated by the time needed to construct the reduced network, namely, O(n 2 k) Recently, a class of networks called row convex generalizing CPA networks was identified and analyzed [25]. 5.4 Augmented PA Networks When we move up the qualitative networks hierarchy from CPA networks to PA networks (allowing also the 6= relation between points) deciding consistency becomes NP hard for discrete domains, and consequently for multiple intervals domains. Proposition 5.25 Deciding ....

P. van Beek and R. Dechter. On the minimality and global consistency of row-convex constraint networks. J. ACM, 42(3):543--561, 1995.

....network which is path consistent, if there exists an indexing of the variables and an ordering of the domain values such that constraints are row convex from the low to the high index variables, then a solution can be obtained without backtracking if one exists. Subsequently, van Beek and Dechter [108] generalized the result to any t ary network which is strongly 2(t 1) 1 consistent. We will outline van Beek s result. Lemma 2.3. Let F be a nite collection of boolean vectors that are row convex and of equal length such that every pair of vectors have a non zero entry in common. Then, all ....

Peter van Beek and Rina Dechter. On the minimality and global consistency of row-convex constraint networks. J. Assoc. Comput. Mach., 42(3):543-561, 1995.

....called as globally consistent. Definition 8.6 Isothetic rectangles are the rectangles whose sides are parallel to coordinate axes 8.2 Consistency Sometimes certain levels of local consistency is enough to guarantee that the network is globally consistent. Ladkin and Maddux[6] VanBeek and Detcher [12] propose several results of determining the value of k such that k consistency implies global consistency. However, the earlier results are restricted to the discrete domains where all possible elements of D i are enumerated. We prove below a result for continuous domain. Report No. 116, July 1997 ....

....are relative ordering among the end points. It is shown that it has restricted expressibility as all interval formulas cannot be expressed as end point formula. We are restricting to disjunctions of relations) 9. 3 SIA Class The next level of generalisation is called Pointizable Algebra[12], Which is also termed as SIA class. Vilain and Kautz show that a restricted class of IA networks, denoted here as SIA networks, can be translated without loss of information into PA networks. Ladkin and Maddux [6] coined the term pointisable relations for those interval relations in IA that can ....

P Van Beek and R Detcher, On the minimality and global consistency of Row-Convex Constraint Network, JACM, 1995, Vol 42, 543-561.

....and GAC4 are even more important than on their binary versions, so that CN can only be applied on ternary constraints and very small domains, and GAC4 on very tight constraints, where the number of allowed tuples of values is very small. Finally, we can point out the work of van Beek and Dechter [ van Beek and Dechter, 1995 ] who proposed another definition for arc consistency on non binary constraint networks, namely relational arc consistency. This definition is much stronger than the classical one since for each constraint it requires global consistency on the underlying subnetwork (i.e. the network involving ....

P. van Beek and R. Dechter. On the minimality and global consistency of row-convex constraint networks. Journal of the ACM, 42(3):543--561, 1995.

....say that a problem is globally solved if it is consistent, and if there is a known ordering of the variables along which solutions can be assembled without encountering deadends. An algorithm globally solves a problem if it generates a globally solved network. 3. 2 Relation based consistency In [25], we extended the notions of arc and path consistency to non binary relations, and used it to specify an alternative condition under which row convex non binary networks are globally consistent. The new local consistency conditions were called relational arc and path consistency. In [24] we ....

....[24, 23] Van Hentenryck, Deville, and Teng [26] show that arc consistency is sufficient to test whether a network is satisfiable if the relations are from a restricted class of functional and monotone constraints. These properties were generalized recently to the property of row convexity [25]. Finally, for work that falls into both classes, Dechter and Pearl [9] present effective procedures for determining whether a constraint network can be formulated as a causal theory and thus a solution can be found without backtracking. Whether a constraint network can be so formulated depends on ....

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P. van Beek and R. Dechter. On the minimality and global consistency of rowconvex constraint networks. To appear in J. ACM, 1995.

.... has been studied empirically (e.g. 10, 7, 17, 18, 19, 28] Second, much previous work has identified conditions for when a certain level of local consistency is sufficient to guarantee a network is globally consistent or to guarantee a solution can be found in a backtrack free manner (e.g. [9, 11, 15, 16, 25, 33]) In this paper, we identify two new complementary properties on the restrictiveness of the constraints in a network constraint tightness and constraint looseness and we show their usefulness for estimating the level of local consistency needed to ensure global consistency, and for estimating ....

....not 3 consistent. For example, given the consistent placement of two queens shown in Figure 1a, there is no way to place a queen in the third column that is consistent with the previously placed queens. Similarly the network is not 4 consistent (see Figure 1b) 3. 2 Relation based consistency In [33], we extended the notions of arc and path consistency to non binary relations, and used it to specify an alternative condition under which row convex non binary networks are globally consistent. The new local consistency conditions were called relational arc and path consistency. We now ....

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P. van Beek and R. Dechter. On the minimality and global consistency of row-convex constraint networks. Accepted for publication in J. ACM, 1994.

.... two ways has been studied empirically (e.g. 4, 6, 13, 14, 23] Second, much previous work has identified conditions for when a certain level of local consistency is sufficient to guarantee that a network is globally consistent or that a solution can be found in a backtrack free manner (e.g. [5, 7, 11, 12, 21, 29]) In this paper, we identify two new complementary properties on the restrictiveness of the constraints in a network constraint tightness and constraint looseness and we show their usefulness for estimating the level of local consistency needed to ensure global consistency, and for estimating ....

....enforced if we want to ensure that a solution can be found in a backtrack free manner. Two definitions of local consistency are employed in characterizing the conditions: the traditional variable based notion and a recently introduced definition of local consistency called relational consistency [9, 29]. 2 Definitions and Preliminaries A constraint network R is a set of n variables fx 1 ; xng; a domain D i of possible values for each variable x i , 1 i n; and a set of t constraint relations fRS1 ; RS t g, where S i f1; ng, 1 i t. Each constraint relation R S i , ....

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P. van Beek and R. Dechter. On the minimality and global consistency of row-convex constraint networks. J. ACM, 42(3):543--561, 1995.

....1 ] 2 RS1 and t[S 2 ] 2 RS2 . Two properties of constraint networks that arise later in the paper are domain size and row convexity. Definition5 (k valued domains) A network of constraints is k valued if the domain sizes of all variables are bounded by k. Definition6 (row convex constraints ([38]) A binary constraint R on a set fx 1 ; x 2 g of variables with associated domains D 1 and D 2 , is row convex if there exists an ordering of D 2 such that, for every a 1 2 D 1 , the set fx 2 j (a 1 ; x 2 ) 2 Rg can be ordered such that the elements appear consecutively in the ordering of D 2 ....

....to the given ordering. An algorithm globally solves a problem if it generates a globally solved network. A globally solved representation is a useful representation of all solutions whenever such a representation is more compact than the set of all solutions. 3. 2 Relation based consistency In [38], we extended the notions of arc and path consistency to non binary relations, and used it to specify an alternative condition under which row convex non binary networks are globally consistent. The new local consistency conditions were called relational arc and path consistency. In [37] we ....

[Article contains additional citation context not shown here]

P. van Beek and R. Dechter. On the minimality and global consistency of rowconvex constraint networks. Journal of the ACM, 42:543-561, 1995.

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