Solving Hierarchies of Finite-Domain Constraints1 2

Solving Hierarchies of Finite-Domain Constraints1 2 Armin Wolf GMD { German National Research Center for Information Technology GMD FIRST, Rudower Chaussee 5, D-12489 Berlin, Germany Phone: +49 30 63 92-18 64 Fax: +49 30 63 92-18 05 e-mail: www:

1

http://www.first.gmd.de

The research in this paper is a result of the projects WISPRO and VERMEIL

funded by the

nology 2

[email protected]

German Federal Ministry for Education, Science, Research, and Tech-

(BMBF) under grant no. 01 IW 206 and no. 01 IN 507 B.

This work will be published in a special issue on Non-Standard Constraint Pro-

cessing in the

Journal on Experimental and Theoretical Arti cial Intelligence , 1997.

Abstract In the past we presented an algorithm to solve ordered constraint hierarchies based on a non-trivial error function using standard constraint satisfaction techniques. We extended this previous work and herewith present a new method to transform hierarchies of inequalities over the integers based on global comparators into equivalent ordered constraint hierarchies. The correctness of this method is proven. Using the results of our previous work, we present another method transforming the resulting ordered constraint hierarchies into ordinary constraint systems. These systems of algebraic equalities and inequalities over the integer domain are soluble with available nite-domain constraint solvers. At last we propose some modi cations and simpli cations of the considered constraint hierarchies improving the search for solutions and being useful in practical applications like job-shop scheduling. The modi cations are the combinations of consecutive hierarchy levels in one level where the constraints are considered with dierent priorities/weights. The simpli cations are incomplete search strategies resulting in run-time improvements and in sub-optimal solutions, too.

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1 Introduction Wide areas in which constraint systems are successfully applied, are computer graphics, especially automatic layout generation [Borning and Duisberg, 1986, Newberry-Paulisch and Tichy, 1990], con guration, and planning, for example assembling of technical systems [Mittal and Falkenhainer, 1990, Stumptner et al., 1994], and scheduling [Caseau and Laburthe, 1994, Goltz, 1995]. In practical applications, we often are confronted with over-constrained problem speci cations which are unsolvable. However, a solution in the form of a layout, plan, schedule, and so on has to be generated. One possibility to solve this con ict is the introduction of constraint hierarchies [Wilson and Borning, 1993]. Constraint hierarchies consist of required constraints which have to be satis ed, and non-required constraints which only have to be satis ed as well as possible.

2 State-of-the-Art The formal foundations of constraint hierarchies are presented in [Borning et al., 1987]. Further developments where made by Molly Ann Wilson in her PhD thesis [Wilson, 1993], especially the extension of constraint logic programming (s.a. [Wilson and Borning, 1993]). For constraint hierarchies based on the trivial error function that returns 0 if the considered constraint is satis ed and 1 if it is not (cf. [Wilson, 1993], page 7), a lot of algorithms to solve such hierarchies are known [Sannella, 1994, Satoh and Aiba, 1991, Tsusumi, 1994]. The rst solution uses method graphs, the second one is a model theoretic approach to the problem, and the third one is based on deductive database techniques. The only algorithm known to the author for constraint hierarchies based on a non-trivial error function { we call an error function non-trivial if its codomain is in nite { is based on the constraint solver DeltaStar [Freeman-Benson et al., 1992] and the Simplex algorithm for linear constraints over the reals. The basics of the algorithm are sketched in Wilson's PhD thesis. Another algorithm for ordered constraint hierarchies based on a non-trivial error function and consisting of algebraic nite domain constraints was presented in [Wolf, 1996]. The use of constraint hierarchies is one possibility to formalize overconstrained systems. The evaluation of solutions of constraint hierarchies requires non-standard constraint processing methods. Both, over-constrained systems, and non-standard constraint processing, are fairly new areas of research [Hower and Ruttkay, 1996, Jampel et al., 1996a]. Besides constraint hierarchies, there are other theoretical approaches to specifying and solving overconstrained problems using non-standard constraint processing techniques. The developed theories have in common that there is some kind of \softness" or

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\fuzziness" of constraints, and some \measures" for these properties, for example error and membership values, priorities, and so on. An overview of the dierent approaches is presented in [Jampel, 1996a]. Recent works in this area deal with a uniform theory subsuming dierent approaches [Bistarelli et al., 1996, Jampel et al., 1996b], a compositional theory of constraint hierarchies [Jampel, 1996b] and their incremental solving [Menezes and Barahona, 1995, Holzbaur et al., 1996].

3 De nitions In this section we give a brief overview of the constraint hierarchy theory presented in [Borning et al., 1987]. For more details [Wilson, 1993] is recommended. A constraint hierarchy H consists of a nite sequence of hierarchy levels H ; H ; : : : ; Hn. Each hierarchy level is a nite sequence of constraints (c ; : : : ; ck ). All constraints at level 0 are required constraints. They have to be satis ed. H is the sequence of the strongest non-required constraints. After considering the constraints at level 0 they have to be satis ed as well as possible. Then the non-required constraints at level 2 are considered and so forth. A valuation is a substitution mapping variables to values of the variables' domains. The application of a valuation  to a constraint c is c. In the following we assume that the valuation  maps (at least) the variables in the considered constraints to values. Given a valuation , a variable x, and a set of variables V we write jx and jV for the valuations having the restricted domains fxg and V , respectively. The application of a valuation  to a sequence of constraints C = (c ; : : : ; ck ) is de ned as C := (c ; : : : ; ck ). An error function e maps any valuated constraint c to a non-negative real value. It holds that e(c) = 0 if and only if the valuation  satis es the constraint c. A generalization of an error function e is a function E mapping valuated sequences of constraints to real-valued vectors. For any sequence of constraints C = (c ; : : : ; ck ) and any valuation  it is de ned as E (C) := [e(c ); : : : ; e(ck )] The error sequence of a constraint hierarchy H = H ; H ; : : : ; Hn with respect to a generalized error function E , and a valuation  is the sequence [E (H ); : : : ; E (Hn)]. An error combining function g \combines" the individual errors e(c ); : : : ; e(ck ) at each hierarchy level Hi to the \error" g(E (Hi)). If the individual errors are really combined to one value, we call the combining function global and the value g(E (Hi)) the error (of the valuation ) at level Hi. A generalization of an error combining function g is a function G de ned as G([E (H ); : : : ; E (Hn)]) := [g(E (H )); : : : ; g(E (Hn))]: 0

1

1

1

1

1

1

1

0

1

1

1

1

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The result [g(E (H )); : : : ; g(E (Hn))] is called a combined error sequence , and represents the \total error" of a valuation  in a hierarchy H . Dierent valuations of a hierarchy are compared with respect to their errors. Therefore, the combined error sequences are compared lexicographically. This requires two binary relations g and 1 result in simpli ed evaluations and sub-optimal solutions. Only c = 1 guarantees to obtain optimal solutions. The complexity of the evaluation of the representatives depends on the number of hierarchy levels. We propose a combination of consecutive hierarchy levels in one level respecting constraints of dierent levels with dierent weights. Replacing the hierarchy levels p; : : : ; q by the hierarchy level p, renaming the variables zp; : : : ; zq to zp, and replacing the constraints ij := (tji ? zj  bji ) for i = 1; : : : ; kj and j = p; : : : ; q in H 0 by ij := (tji ? cj
zp  bji ) for i = 1; : : : ; kj and j = p; : : : ; q 1

1

1

0

1

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results in such a combination if we assume that the combining function g is de ned as g(x ; : : : ; xm ) := maxmi xi (for other global comparators the replacement has to be accomplished analogously). We also propose either constant, linearly, or exponentially growing values cj (w.r.t. j ). It is obvious that the error of the resulting hierarchy of a valuation  at level Hp00 which represents the levels (Hp; : : : ; Hq ) is 1

=1

(

)

e(ij ) j i = 1; : : : ; k and j = p; : : : ; q : j p cj Reduced complexity is one advantage of this combination; another advantage is the consideration of constraints at lower levels during the minimization of the global errors at higher levels. g(E (H 00)) := max

9 Conclusion and Future Work In this paper we present a new method to transform algebraic constraint hierarchies based on a global comparator into ordered hierarchies. We also prove the correctness of the transformation. The presented methods are implemented in ECLi PSe, the constraint logic programming system of the ECRC [ECR, 1995b] using the build-in nite-domain constraint solver [ECR, 1995a]. We used the prototype implementation to solve some small over-constrained scheduling problems. We are now investigating some eorts to formalize the \softness" of composite and global constraints, for example cumulative constraints [Aggoun and Beldiceanu, 1993], which are necessary to model nite capacity scheduling problems. Further we try to improve the optimizations which are necessary to evaluate the representatives. Incremental Search [van Hentenryck and le Provost, 1991] seems to be a good candidate to replace the used branch-and-bound algorithm. In future we plan to apply the given results to model and solve overconstrained real-world scheduling and con guration problems. To model these problems adequately it seems necessary to use dierent comparators in the same constraint hierarchy. Having the adapted transformations for the global comparators in mind, it is easy to choose dierent comparators for dierent levels. In this application context we want to integrate a rule-based component which supports the modeling, especially the assembly of the hierarchies and some heuristics reducing the time complexity.

Acknowledgments

This work originates from the German National Research Center for Information Technology (GMD), Research Institute for Computer Architecture and Software

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Technology (FIRST) and was supported by the German Federal Ministry for Education, Science, Research, and Technology (BMBF). Many thanks to the members of the constraint logic programming group at FIRST for fruitful discussions helping to improve this paper.

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