# Inequality-sum : a global constraint capturing the objective function

• Volume: 39, Issue: 2, page 123-139
• ISSN: 0399-0559

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## Abstract

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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.

## How to cite

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Régin, Jean-Charles, and Rueher, Michel. "Inequality-sum : a global constraint capturing the objective function." RAIRO - Operations Research - Recherche Opérationnelle 39.2 (2005): 123-139. <http://eudml.org/doc/245189>.

@article{Régin2005,
abstract = {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.},
author = {Régin, Jean-Charles, Rueher, Michel},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
language = {eng},
number = {2},
pages = {123-139},
publisher = {EDP-Sciences},
title = {Inequality-sum : a global constraint capturing the objective function},
url = {http://eudml.org/doc/245189},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Régin, Jean-Charles
AU - Rueher, Michel
TI - Inequality-sum : a global constraint capturing the objective function
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 2
SP - 123
EP - 139
AB - 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.
LA - eng
UR - http://eudml.org/doc/245189
ER -

## References

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