Inequality-sum : a global constraint capturing the objective function

Régin, Jean-Charles; Rueher, Michel

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.

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

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

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

[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

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

[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

[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

[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

[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 | Zbl

[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