Inequality-sum : a global constraint capturing the objective function
RAIRO - Operations Research - Recherche Opérationnelle, Tome 39 (2005) no. 2, pp. 123-139.

This paper introduces a new method to prune the domains of the variables in constrained optimization problems where the objective function is defined by a sum $y=\Sigma {x}_{i}$, and where the integer variables ${x}_{i}$ are subject to difference constraints of the form ${x}_{j}-{x}_{i}\le c$. An important application area where such problems occur is deterministic scheduling with the mean flow time as optimality criteria. This new constraint is also more general than a sum constraint defined on a set of ordered variables. Classical approaches perform a local consistency filtering after each reduction of the bound of $y$. The drawback of these approaches comes from the fact that the constraints are handled independently. We introduce here a global constraint that enables to tackle simultaneously the whole constraint system, and thus, yields a more effective pruning of the domains of the ${x}_{i}$ when the bounds of $y$ are reduced. An efficient algorithm, derived from Dijkstra’s shortest path algorithm, is introduced to achieve interval consistency on this global constraint.

DOI : https://doi.org/10.1051/ro:2005007
Classification : 65K05,  90C35
@article{RO_2005__39_2_123_0,
author = {R\'egin, Jean-Charles and Rueher, Michel},
title = {Inequality-sum : a global constraint capturing the objective function},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {123--139},
publisher = {EDP-Sciences},
volume = {39},
number = {2},
year = {2005},
doi = {10.1051/ro:2005007},
zbl = {1104.90051},
mrnumber = {2181795},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro:2005007/}
}
Régin, Jean-Charles; Rueher, Michel. Inequality-sum : a global constraint capturing the objective function. RAIRO - Operations Research - Recherche Opérationnelle, Tome 39 (2005) no. 2, pp. 123-139. doi : 10.1051/ro:2005007. http://www.numdam.org/articles/10.1051/ro:2005007/

[1] R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network Flows. Prentice Hall (1993). | MR 1205775

[2] N. Beldiceanu and E. Contejean, Introducing global constraints in chip. J. Math. Comput. Model. 20 (1994) 97-123. | Zbl 0816.68048

[3] J. Blazewicz, K. Ecker, G. Schmidt and J. Weglarz, Scheduling in Computer and Manufacturing Systems. Springer-Verlag (1993). | Zbl 0767.90033

[4] J. Carlier and E. Pinson, A practical use of jackson's preemptive schedule for solving the job-shop problem. Ann. Oper. Res. 26 (1990) 269-287. | Zbl 0709.90061

[5] R. Dechter, I. Meiri and J. Pearl, Temporal constraint networks. Artif. Intell. 49 (1991) 61-95. | Zbl 0737.68070

[6] M. Dror, W. Kubiak and P. Dell'Olmo, Scheduling chains to minimize mean flow time. Inform. Process. Lett. 61 (1997) 297-301.

[7] P. Van Hentenryck and Y. Deville, The cardinality operator: A new logical connective for constraint logic programming, in Proc. of ICLP'91 (1991) 745-759.

[8] P. Van Hentenryck, V. Saraswat and Y. Deville, Design, implementation, and evaluation of the constraint language cc(FD). J. Logic Program. 37 (1998) 139-164. | Zbl 0920.68026

[9] M. Rueher and J.-C. Régin, A global constraint combining a sum constraint and difference constraints, in Proc. of CP'2000, Sixth International Conference on Principles and Practice of Constraint Programming. Singapore, Springer-Verlag. Lect. Notes Comput. Sci. 1894 (2000) 384-395. | Zbl 1044.68794

[10] C.E. Leiserson and J.B. Saxe, A mixed-integer linear programming problem which is efficiently solvable. J. Algorithms 9 (1988) 114-128. | Zbl 0649.90077

[11] J.-C. Régin, A filtering algorithm for constraints of difference in CSPs, in Proc. of AAAI-94. Seattle, Washington (1994) 362-367.

[12] J.-C. Régin, Generalized arc consistency for global cardinality constraint, in Proc. of AAAI-96, Portland, Oregon (1996) 209-215.

[13] H. Simonis, Problem classification scheme for finite domain constraint solving, in CP96, Workshop on Constraint Programming Applications: An Inventory and Taxonomy, Cambridge, MA, USA (1996) 1-26.

[14] R.E. Tarjan, Data Structures and Network Algorithms. CBMS 44 SIAM (1983). | MR 826534 | Zbl 0584.68077

[15] P. Van Hentenryck, Y. Deville and C.-M. Teng, A generic arc-consistency algorithm and its specializations. Artif. Intell. 57 (October 1992) 291-321. | Zbl 0763.68059