Abstract
A constraint satisfaction problem CSP consists of assigning values to variables which are subject to a set of constraints. This problem can be modelized as a 0–1 quadratic knapsack problem. We show that a CSP has a solution if and only if the value of the optimization problem is null. A branch-and-bound method exploiting this representation is presented. At each node, a lower bound given naturally by the value of dual problem may be improved by a new Lagrangean heuristic. An upper bound is computed by satisfying the quadratic constraint. The theoritical results presented are used for filtring the associated subproblem and for detecting quickly a solution or a failure. The simulation results assess the effectiveness of the theoretical results shown in this paper.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Balas E., Mazzola J.B. (1984), “Nonlinear 0-1 programming linearisation techniques”, Mathematical programming 30,1–21.
Barahona F., Jüngler M. and Reinelt G. (1989), “Experiments in quadratic 0–1 program-ming”, Mathematical programming 44, 127–137.
Bessière C.(1994),“Arc-consistency and arc-consistency again”, Research note,Vol. 65
Billionnet A., Sutter A. (1992), “Persistency in quadratic 0–1 optimization”, Mathematical programming 54, 115–119.
Carter M.W. (1984), “The indefinite 0–1 Quadratic problem”, Discrete Applied Mathematics, 7, 23–44.
Crania Y. (1989), “Recognition problems for special classes of polynomials in 0–1 variables”, Mathematical programming 44, 127–137.
Ettaouil M. (1994), “Satisfaction de contraintes non linéaire en vatiables 0–1 et outils de la programmation quadratique”, Thèse d'Université, Université Paris 13.
Ettaouil M., Plateau G.(1996), “An exact algorithm for the nonlinear 0–1 constraint Satisfaction problem”, Rapport de recherche LIPN 96-10, Université Paris 13.
Geoffrion A. M. (1974), “The Lagrangian relaxation for integer programming”, Mathematical Programming Study 2, 82–114.
Glover F., Woolsey E. (1974), “Converting the 0–1 polynomial programming problem to 0–1 linear program”, Operations research 22, 180–182.
Kalanhari B. and Bagchi A. (1990), “An algorithm for quadratic 0–1 programs”, Naval Research Logistic, 37, 527–538.
Martello S., Toth P. (1977), “An upper bound for the 0–1 Knapsack problem and a branch and bound algorithm”, European Journal of Operations Research 1,169–175.
Michelon P. et Maculan N. (1991), “Lagrangian methods for 0–1 quadratic problems”, Discrete Applied Mathematics 42, 257–269.
Minoux M. (1983), “Programmation Mathématique ”, Tome 2 Dunod.
Mohr R. and Henderson T. C.(1986), “Arc and path consistency revisid” Artificial Intelligence 28-2.
Montanari U. (1974), “Networks of constraints: Fundamental proprieties and applications to picture processing”, Information Sciences, Vol. 7 N°2, 95–132.
Pardalos P.M., Rogers G.P. (1990), “Computational Aspects of a Branch and Bound algorithm for quadratic 0–1 programming”, Computing, 45, 131–144.
Watters L.J. (1967), “Reduction of integer polynomial programming problems to zero-one linear programming problems”, Operations Research 15, 1171–1174.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ettaouil, M. (1997). A 0–1 quadratic knapsack problem for modelizing and solving the constraint satisfaction problems. In: Coasta, E., Cardoso, A. (eds) Progress in Artificial Intelligence. EPIA 1997. Lecture Notes in Computer Science, vol 1323. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023911
Download citation
DOI: https://doi.org/10.1007/BFb0023911
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63586-4
Online ISBN: 978-3-540-69605-6
eBook Packages: Springer Book Archive