A NOTE ON DUAL APPROXIMATION ALGORITHMS FOR CLASS CONSTRAINED BIN PACKING PROBLEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In this paper we present a dual approximation scheme for the class constrained shelf bin packing problem. In this problem, we are given bins of capacity 1, and n items of Q different classes, each item e with class c(e) and size s(e). The problem is to pack the items into bins, such that two items of different classes packed in a same bin must be in different shelves. Items in a same shelf are packed consecutively. Moreover, items in consecutive shelves must be separated by shelf divisors of size d. In a shelf bin packing problem, we have to obtain a shelf packing such that the total size of items and shelf divisors in any bin is at most 1. A dual approximation scheme must obtain a shelf packing of all items into N bins, such that, the total size of all items and shelf divisors packed in any bin is at most 1 + epsilon for a given epsilon > 0 and N is the number of bins used in an optimum shelf bin packing problem. Shelf divisors are used to avoid contact between items of different classes and can hold a set of items until a maximum given weight. We also present a dual approximation scheme for the class constrained bin packing problem. In this problem, there is no use of shelf divisors, but each bin uses at most C different classes.432239248Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Faepex [31608]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2008/01490-3]Faepex [31608]CNPq [478470/06-1, 472504/07-0, 306624/07-9

Passage à l'échelle pour les contraintes d'ordonnancement multi-ressources

La programmation par contraintes est une approche régulièrement utilisée pour résoudre des problèmes combinatoires d origines diverses. Dans cette thèse nous nous focalisons sur les problèmes d ordonnancement cumulatif. Un problème d ordonnancement consiste à déterminer les dates de débuts et de fins d un ensemble de tâches, tout en respectant certaines contraintes de capacité et de précédence. Les contraintes de capacité concernent aussi bien des contraintes cumulatives classiques où l on restreint la somme des hauteurs des tâches intersectant un instant donné, que des contraintes cumulatives colorées où l on restreint le nombre maximum de couleurs distinctes prises par les tâches. Un des objectifs récemment identifiés pour la programmation par contraintes est de traiter des problèmes de grandes tailles, habituellement résolus à l aide d algorithmes dédiés et de métaheuristiques. Par exemple, l utilisation croissante de centres de données virtualisés laisse apparaitre des problèmes d ordonnancement et de placement multi-dimensionnels de plusieurs milliers de tâches. Pour atteindre cet objectif, nous utilisons l idée de balayage synchronisé considérant simultanément une conjonction de contraintes cumulative et des précédences, ce qui nous permet d accélérer la convergence au point fixe. De plus, de ces algorithmes de filtrage nous dérivons des procédures gloutonnes qui peuvent être appelées à chaque nœud de l arbre de recherche pour tenter de trouver plus rapidement une solution au problème. Cette approche permet de traiter des problèmes impliquant plus d un million de tâches et 64 ressources cumulatives. Ces algorithmes ont été implémentés dans les solveurs de contraintes Choco et SICStus, et évalués sur divers problèmes déplacement et d ordonnancement.Mots-clés : Programmation par contraintes, ordonnancement, cumulatif, passage à l échelle, point fixe, contraintes de ressources multidimensionelles, balayage synchronisé.Constraint programming is an approach often used to solve combinatorial problems in different application areas. In this thesis we focus on the cumulative scheduling problems. A scheduling problem is to determine the starting dates of a set of tasks while respecting capacity and precedence constraints. Capacity constraints affect both conventional cumulative constraints where the sum of the heights of tasks intersecting a given time point is limited, and colored cumulative constraints where the number of distinct colors assigned to the tasks intersecting a given time point is limited. A newly identified challenge for constraint programming is to deal with large problems, usually solved by dedicated algorithms and metaheuristics. For example, the increasing use of virtualized datacenters leads to multi dimensional placement problems of thousand of jobs. Scalability is achieved by using a synchronized sweep algorithm over the different cumulative and precedence constraints that allows to speed up convergence to the fix point. In addition, from these filtering algorithms we derive greedy procedures that can be called at each node of the search tree to find a solution more quickly. This approach allows to deal with scheduling problems involving more than one million jobs and 64 cumulative resources. These algorithms have been implemented within Choco and SICStussolvers and evaluated on a variety of placement and scheduling problems.NANTES-ENS Mines (441092314) / SudocSudocFranceF

The Class Constrained Bin Packing Problem With Applications To Video-on-demand

In this paper we present approximation results for the class constrained bin packing problem that has applications to Video-on-Demand Systems. In this problem we are given bins of capacity B with C compartments, and n items of Q different classes. The problem is to pack the items into the minimum number of bins, where each bin contains items of at most C different classes and has total items size at most B. We present several approximation algorithms for off-line and online versions of the problem. © Springer-Verlag Berlin Heidelberg 2006.4112 LNCS439448Coffman Jr., E.G., Garey, M.R., Johnson, D.S., Approximation algorithms for bin packing: A survey (1997) Approximation Algorithms for NP-hard Problems, pp. 46-93. , D. Hochbaum, editor, chapter 2. PWSDawande, M., Kalagnanam, J., Sethuraman, J., Variable sized bin packing with color constraints (1998) Technical Report, , IBM, T.J. Watson Research Center, NYDawande, M., Kalagnanam, J., Sethuraman, J., Variable sized bin packing with color constraints (2001) Electronic Notes in Dicrete Mathematics, 7. , Proceedings of Graco 2001De La Vega, W.F., Lueker, G.S., Bin packing can be solved within 1 + ε in linear time (1981) Combinatorica, 1 (4), pp. 349-355Golubchik, L., Khanna, S., Khuller, S., Thurimella, R., Zhu, A., Approximation algorithms for data placement on parallel disks (2000) Proceedings of SODA, pp. 223-232Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L., Worst-case performance bounds for simple one-dimensional packing algorithms (1974) SIAM Journal on Computing, 3, pp. 299-325Kashyap, S.R., Khuller, S., Algorithms for non-uniform size data placement on parallel disks (2003) Lecture Notes in Computer Science, 2914, pp. 265-276. , Proceedings of FSTTCSShachnai, H., Tamir, T., On two class-constrained versions of the multiple knapsack problem (2001) Algorithmica, 29, pp. 442-467Shachnai, H., Tamir, T., Polynomial time approximation schemes for class-constrained packing problems (2001) Journal of Scheduling, 4 (6), pp. 313-338Shachnai, H., Tamir, T., Approximation schemes for generalized 2-dimensional vector packing with application to data placement (2003) Lecture Notes in Computer Science, 2764, pp. 165-177. , Proceedings of 6th RANDOM-APPROXShachnai, H., Tamir, T., Tight bounds for online class-constrained packing (2004) Theoretical Computer Science, 321 (1), pp. 103-12

A One-dimensional Bin Packing Problem With Shelf Divisions

Given bins of size B, non-negative values d and Δ, and a list L of items, each item e ∈ L with size se and class ce, we define a shelf as a subset of items packed inside a bin with total item sizes at most Δ such that all items in this shelf have the same class. Two subsequent shelves must be separated by a shelf division of size d. The size of a shelf is the total size of its items plus the size of the shelf division. The class constrained shelf bin packing problem (CCSBP) is to pack the items of L into the minimum number of bins, such that the items are divided into shelves and the total size of the shelves in a bin is at most B. We present hybrid algorithms based on the First Fit (Decreasing) and Best Fit (Decreasing) algorithms, and an APTAS for the problem CCSBP when the number of different classes is bounded by a constant C. © 2007 Elsevier B.V. All rights reserved.156710831096Coffman Jr., E.G., Garey, M.R., Johnson, D.S., Approximation algorithms for bin packing: a survey (1997) Approximation Algorithms for NP-hard Problems, pp. 46-93. , Hochbaum D. (Ed), PWS (Chapter 2)M. Dawande, J. Kalagnanam, J. Sethuranam, Variable sized bin packing with color constraints, First Brazilian Symposium on Graph, Algorithms and Combinatorics, Electronic Notes in Discrete Mathematics, vol. 7, 2001, pp. 1-4Fernandez de la Vega, W., Lueker, G.S., Bin packing can be solved within 1 + ε{lunate} in linear time (1981) Combinatorica, 1 (4), pp. 349-355Ferreira, J.S., Neves, M.A., Fonseca e Castro, P., A two-phase roll cutting problem (1990) European J. Oper. Res., 44 (2), pp. 185-196Hoto, R., Arenales, M., Maculan, N., The one dimensional compartmentalized cutting stock problem: a case study (2007) European J. Oper. Res., 183 (3), pp. 1183-1195Marques, F.P., Arenales, M., The constrained compartmentalized knapsack problem (2007) Comput. Oper. Res., 34 (7), pp. 2109-2129Shachnai, H., Tamir, T., Polynomial time approximation schemes for class-constrained packing problems (2001) J. Scheduling, 4 (6), pp. 313-338Shachnai, H., Tamir, T., Tight bounds for online class-constrained packing (2004) Theoret. Comput. Sci., 321 (1), pp. 103-123Simchi-Levi, D., New worst-case results for the bin-packing problem (1994) Naval Res. Logistics, 41, pp. 579-585Vazirani, V., (2001) Approximation Algorithms, , Springer, BerlinXavier, E.C., Miyazawa, F.K., Approximation schemes for knapsack problems with shelf divisions (2006) Theoret. Comput. Sci., 352 (1-3), pp. 71-8