P. Hubbe and E. Freuder (1992). An efficient cross-product representation of the constraint satisfaction problem search space. In Proc. of AAAI-92, pages 421--427, San Jose, CA.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... possible configurations; ffl in any case, it may be advantageous to produce a solution set, rather than a single solution; in [Les93] the authors present an algorithm which produces particular solution sets: cross products of sub domains of the variables; this work is clearly related to [Fre91, HF92] ffl it is finally always possible to wait for the changes and to react when they occur; the problem becomes dynamic (see 3.3) 3.3 Dynamicity This problem, formalized in [DD88] is currently the subject of numerous studies. We can distinguish two approaches: ffl maintaining, change after ....

Paul D. Hubbe and Eugene C. Freuder, An efficient cross-product representation of the constraint satisfaction problem search space, Proc. of AAAI-92 (San Jose, CA), 1992, pp. 421--427.

.... PARADIGM 82 [Mackworth, 1987] Mackworth, 1992] Mackworth and Freuder, 1993] Mackworth and Freuder, 1985] Mackworth, 1977] Dechter and Pearl, 1987] Dechter and Pearl, 1989] Dechter and Meiri, 1989] Dechter and Dechter, 1987] Dechter, 1990a] Dechter, 1990b] Dechter, 1992] [Hubbe and Freuder, 1992], Freuder and Mackworth, 1992] Freuder and Wallace, 1992] Freuder, 1982] and [Freuder, 1991] In this section I intend to provide a brief description of constraint satisfaction problems, and of how they are typically solved with a combination of heuristic search and constraint propagation ....

Paul Hubbe and Eugene Freuder. An efficient cross product representation of the constraint satisfaction problem search space. In Proceedings of the 10th AAAI, pages 421--427, 1992.

....In this paper we address CSPs with many solutions which are to be inspected to select the most suitable one. Structuring this space of solutions means constructing a more efficient representation which simplifies such a selection. An example of such a structure is the crossproduct representation [ Hubbe and Freuder, 1992 ] Interchangeability [ Freuder, 1991 ] is an important concept for structuring CSPs. It involves replacing equivalent values by single meta values, thus collapsing solutions with either value to a single one. Crossproduct representations can be constructed from interchangeabilities. The ....

....are allowed by the constraint between the two nodes and these tuples correspond to the solutions for the CSP on this level (Fig. 10) Every meta value in a solution of the resulting CSP can be replaced by the set of values it represents. In this way one obtains the cross product representation [ Hubbe and Freuder, 1992 ] of solutions for the next lower level. Replacing meta values for each solution in the crossproduct continues until the lowest level is reached. The 210 solutions can be refined to 438 on the next lower level then to 2910 and finally to the 5366 solutions of the original CSP. It is worth to ....

P.D. Hubbe and E.C. Freuder. An efficient Cross Product Representation of the Constraint Satisfaction Search Space. In Proc. of AAAI-92, pages 421--427, San Jose, CA, 1992.

....connexes, ayant 16 variables et des domaines de taille 9. Le but etant d evaluer les difficultes de resolution, une ( tranche ) a taux de satisfiabilite fixe des contraintes (60 ) a ete examinee dans l espace traditionnel (taux de satisfiabilite, densite du graphe) tel qu il est decrit dans [HUB 92] Le nombre de contraintes varie de 15 (CSP arborescent) a 120 (graphe complet) par pas de 7 contraintes. A chaque point, 50 CSP ont ete generes et resolus. Les valuations des contraintes sont tirees aleatoirement, avec une distribution uniforme, dans l ensemble f0:25; 0:50; 0:75; 1g. Ces ....

HUBBE P. D. et FREUDER E. C., An efficient cross-product representation of the constraint satisfaction problem search space. In : Proc. of AAAI-92, p. 421--427, San Jose, CA, 1992.

....problemes dynamiques generes de facon aleatoire. Nous rappelons au lecteur les limitations inherentes a l utilisation de ce type de problemes (possible manque de representativite) 5. 1 Generation Le modele de generation aleatoire utilise est une extension au cas dynamique du modele propose dans [HUB 92] Chaque probleme dynamique est genere avec : un nombre de variables egal a n ; une taille des domaines egale a d, identique pour chacune des variables ; uniquement des contraintes binaires ; DBT DYN(p; C a ; C r ) let be V 1 = ASSIGNED VARIABLES(p) let be V 2 = ....

HUBBE P. et FREUDER E., An Efficient Cross-Product Representation of the Constraint Satisfaction Problem Search Space. In : Proc. of AAAI-92, p. 421--427, San Jose, CA, 1992.

....could be generated with the same set of parameters. One of the consequences of this fact was that a very large number of instances must be solved to predict the behavior of an algorithm with a good statistical validity. Hence, a new generation of instance generators appeared (beginning with [18]) which replaced the probability p 1 to have a constraint between two variables by a fixed number C of constraints [24] In the same way, p 2 can be replaced by a number T of forbidden value pairs [11] In [30] p 1 and p 2 are still used, but they represent proportions 5 The idea of breaking ....

P.D. Hubbe and E.C. Freuder. An efficient cross product representation of the constraint satisfaction problem search space. In Proceedings AAAI'92, pages 421--427, San Jose CA, 1992.

....v i of the minimum on all values 2 d i of the valuation of the assignment fv i = g in this problem. 3. 6 Experimentations The Forward Checking algorithm has been coded and applied to random VCSP generated as follows: a classical random CSP with 16 variables, domains of size 9 is generated as in [12]. A first possibilistic VCSP is obtained by randomly assigning a valuation 1 4 ; 1 2 ; 3 4 or 1 to each constraint. A lexicographic VCSP is then built simply by using the transformation from possibilistic to lexicographic CSP described in section 3.2. This VCSP is a strong refinement of the ....

Paul D. Hubbe and Eugene C. Freuder. An efficient cross-product representation of the constraint satisfaction problem search space. In Proc. of AAAI-92, pages 421--427, San Jose, CA, 1992.

....of working with (prime) implicants as (greatest) lower bounds, we can work with (minimal) allgoods. In CSPs where more than one tuple is allowed per constraint (unlike the usual case for satisfiability) the CSP DP analogue may benefit from using the search space cross product representation of Hubbe and Freuder [1992]. 4 Refinements to Bootstrapped KC Here we discuss several refinements to our basic bootstrapped KC approach. In Section 4.1, we examine the costs and benefits of generating prime implicant covers for exact KC which are irredundant. In Section 4.2, we evaluate a heuristic ordering for the ....

Hubbe, Paul and Freuder, Eugene 1992. An efficient cross product representation of the constraint satisfaction problem search space. In Proceedings of the Tenth National Conference on Artificial Intelligence (AAAI-92). 421--427.

....time algorithms for enforcing arc consistency [Rosenfeld et al. 1976; Snow and Freuder, 1990] 4. 6 Experimentations The Forward Checking algorithm has been coded and applied to random VCSP generated as follows: a classical random CSP with 16 variables, domains of size 9 is generated as in [Hubbe and Freuder, 1992] . A first possibilistic VCSP is obtained by randomly assigning a valuation 1 4 ; 1 2 ; 3 4 or 1 to each constraint. A lexicographic VCSP is then built simply by using the transformationfrom possibilistic to lexicographic CSP described in section 3. This VCSP is a strong refinement of the ....

Paul D. Hubbe and Eugene C. Freuder. An efficient cross-product representation of the constraint satisfaction problem search space. In Proc. of AAAI-92, pages 421-- 427, San Jose, CA, 1992.

....v i of the minimum on all values 2 d i of the valuation of the assignment fv i = g in this problem. 3. 7 Experiments The Forward Checking algorithm has been coded and applied to random VCSP generated as follows: a classical random CSP with 16 variables and domains of size 9 is generated as in [13]. A first possibilistic VCSP is obtained by randomly assigning a valuation 1 4 ; 1 2 ; 3 4 or 1 to each constraint. A lexicographic VCSP is then built simply by using the transformation from possibilistic to lexicographic CSP described in Sect. 3.3. This VCSP is a strong refinement of the ....

P. Hubbe and E. Freuder. An efficient cross-product representation of the constraint satisfaction problem search space. In Proc. of AAAI-92, pages 421--427, San Jose, CA, 1992.

....in a recursive divide and conquer form, alternating local consistency processing with variable domain splitting. The schema formulation helps suggest a variety of NC variations to explore, and clarifies the relationship between NC and FC. 4. backtracking with cross product representation (BT CPR) [Hubbe and Freuder, 1992]: The schema formulation, similar to that used in the original presentation of the algorithm, facilitates viewing backtracking as a degenerate case of backtracking with CPR. This in turn facilitates the demonstration that adding CPR can not increase the number of constraint checks (a standard ....

....the search space after finding solutions ( pretending to fail) they will find all solutions. CPR, a relatively new decomposition, is based on the insight that sets of incomplete solutions may be represented and efficiently processed in a cross product representation. It is proven analytically in [Hubbe and Freuder, 1992] that BT CPR never requires more constraint checks than BT, when searching for all solutions, or proving that none exist. A similar result is obtained for CPR in conjunction with FC. The new decomposition, IDC, is based on the hypothesis that, if P is solvable, either there will be a solution ....

P. Hubbe and E. Freuder. An efficient cross product representation of the constraint satisfaction problem search space. Proceedings of the Tenth National Conference on Artificial Intelligence. 421-427.

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P. Hubbe and E. Freuder (1992). An efficient cross-product representation of the constraint satisfaction problem search space. In Proc. of AAAI-92, pages 421--427, San Jose, CA.

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Hubbe, P. D., et Freuder, E. C. 1992. An efficient cross-product representation of the constraint satisfaction problem search space. Dans Proc. of AAAI'92, 421--427. 9, 22, 23

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P. Hubbe and E. Freuder. An efficient cross-product representation of the constraint satisfaction problem search space. In Proc. of AAAI-92, pages 421--427, San Jose, CA, 1992.

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P. Hubbe and E. Freuder. An Efficient Cross-Product Representation of the Constraint Satisfaction Problem Search Space. In Proc. of AAAI-92, pages 421--427, San Jose, CA, 1992.

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