Recently, many algorithms have been designed to propagate global constraints. Unfortunately, some global constraints, such the AT-MOST-1 constraint and the EXTENDEDGCC are NP-Hard to propagate. Often, these constraints can easily be written as integer linear programs. Using linear relaxations and other techniques developed by the operation research community, we want to efficiently propagate such constraints. We model constraints as integer programs that we relax into linear programs. For each value v in a variable domain dom(x), we create a binary variable xv. The assignment xv= 1 indicates that x = v while xv= 0 indicates that x ≠ v. The binary variables are subject to the following linear program. Ax≤b 0≤xi≤1}P We find a solution to the relaxation using the interior point method. This method always converges to the interior of the solution polytope. Based on this observation, we conclude that if a variable xi subject to l ≤ xi ≤ u is assigned to one of its boundary value I or u, the variable is assigned to this value in any solution. Therefore, if the interior point method assigns variable x v.to 0, we conclude that v does not have a support in dom(x). We studied the consistency enforced by our propagator. We proved that if U is a totally unimodular matrix and that P has the following form, then GAC is enforced on the equations Ux ≤ b1. [U 0 B C] [x y] ≤ [b1 b2] 0 ≤ xi,yj ≤ 1} P Some constraints like the CARDINALITY-MATRIX constraint while written in the form Ax ≥ b have many totally unimodular sub-matrices of A. For each such sub-matrix, GAC is enforced on the corresponding variables. The resulting consistency is stronger than consistencies enforced by existing propagators. © Springer-Verlag Berlin Heidelberg 2005.
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Quimper, C. G., & López-Ortiz, A. (2005). From linear relaxations to global constraint propagation. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 3709 LNCS, p. 867). https://doi.org/10.1007/11564751_105