On-line bin-packing problem : maximizing the number of unused bins

In this paper, we study the on-line version of the bin-packing problem. We analyze the approximation behavior of an on-line bin-packing algorithm under an approximation criterion called differential ratio. We are interested in two types of results : the differential competitivity ratio guaranteed by the on-line algorithm and hardness results that account for the difficulty of the problem and for the quality of the algorithm developed to solve it. In its off-line version, the bin-packing problem, BP, is better approximated in differential framework than in standard one. Our objective is to determine if or not such result exists for the on-line version of BP.On-line algorithm, bin-packing problem, competitivity ratio.

On-line computation and maximum-weighted hereditary subgraph problems

URL des Cahiers : https://halshs.archives-ouvertes.fr/CAHIERS-MSE Voir aussi l'article basé sur ce document de travail paru dans "International Symposium on Algorithms and Computation", ISAAC 2005: Algorithms and Computation pp 433-442Cahiers de la Maison des Sciences Economiques 2006.34 - ISSN 1624-0340In this paper, we study the on-line version of maximum-weighted hereditary subgraph problems. In our on-line model, the final instance-graph (which has n vertices) is revealed in t clusters, 2 ≤ t ≤ n. We first focus on the on-line version of the following problem: finding a maximum-weighted subgraph satisfying some hereditary property. Then, we deal with the particular case of the independent set. For all these problems, we are interested in two types of results: the competitivity ratio guaranteed by the on-line algorithm and hardness results that account for the difficulty of the problems and for the quality of algorithms developed to solve them.Dans ce document, nous commençons par étudier la version on-line du problème du sous-graphe héréditaire de poids maximum, WHG, ci-dessous défini : étant donné un graphe G et une propriété héréditaire, trouver un sous-graphe de G de poids maximum satisfaisant. Ensuite, nous étudierons le cas particulier du problème du stable pondéré. Dans notre modèle on-line, nous supposons que l'instance finale de taille n'est révélée en t étapes (ou paquets), 2 ≤ t ≤ n. Nous analysons le comportement des algorithmes on-line résolvant le problème WHG et déterminons des rapports compétitifs (résultats positifs) et des résultats négatifs. Ces derniers résultats rendent compte aussi bien de la difficulté du problème que de la qualité des algorithmes élaborés pour les résoudre

Coloration avec préférences : complexité, inégalités valides et vérification formelle

National audienceNous nous intéressons à un problème de coloration avec préférences minimale CPM dans les graphes triangulés. Cette étude s'inscrit dans le projet CompCert qui a pour objectif la certification, à l'aide de méthodes formelles, d'un compilateur optimisant du langage C. L'une des optimisations du compilateur certifié est l'allocation des registres du processeur. Optimiser cette allocation de registres revient à résoudre le problème CPM auquel nous nous intéressons. Nous montrons un résultat de complexité concernant CPM et proposons l'amélioration d'une méthode de coupes permettant la résolution de ce problème. Ce travail est une jonction entre la recherche opérationnelle et les méthodes formelles, dans la mesure où nous vérifions formellement par ailleurs la résolution du problème en prouvant correct le développement, hormis la recherche effectuée par le solveur dont la vérification consiste à déterminer a posteriori si la solution proposée est bien correcte

Rewriting integer variables into zero-one variables: some guidelines for the integer quadratic multi-knapsack problem

This paper is concerned with the integer quadratic multidimensional knapsack problem (QMKP) where the objective function is separable. Our objective is to determine which expansion technique of the integer variables is the most appropriate to solve (QMKP) to optimality using the upper bound method proposed by Quadri et al. (2007). To the best of our knowledge the upper bound method previously mentioned is the most effective method in the literature concerning (QMKP). This bound is computed by transforming the initial quadratic problem into a 0–1 equivalent piecewise linear formulation and then by establishing the surrogate problem associated. The linearization method consists in using a direct expansion initially suggested by Glover (1975) of the integer variables and in applying a piecewise interpolation to the separable objective function. As the direct expansion results in an increase of the size of the problem, other expansions techniques may be utilized to reduce the number of 0–1 variables so as to make easier the solution to the linearized problem. We will compare theoretically the use in the upper bound process of the direct expansion (I) employed in Quadri et al. (2007) with two other basic expansions, namely: (II) a direct expansion with additional constraints and (III) a binary expansion. We show that expansion (II) provides a bound which value is equal to the one computed by Quadri et al (2007). Conversely, we provide the proof of the non applicability of expansion (III) in the upper bound method. More specifically, we will show that if (III) is used to rewrite the integer variables into 0–1 variables then a linear interpolation can not be applied to transform (QMKP) into an equivalent 0–1 piecewise linear problem.ou

A roof linearization algorithm to obtain a tight upper bound for integer nonseparable quadratic programming

We study in this paper a general case of integer quadratic multi-knapsackproblem (QMKP) where the objective function is non separable. An upperbound method is proposed for (QMKP) which is computed via two steps.The rst stage aims to rewrite (QMKP) into an equivalent mixed integerquadratic program (QPxy) where the objective function is separable, usingGauss decomposition of the quadratic terms matrix. We then suggest anoriginal technique, we call roof linearization, to linearize (QPxy) so as to obtain a mixed linear program which optimal value provides an upper bound for (QPxy) and consequently for (QMKP). Preliminary computational ex-periments are conducted so as to assess that the proposed algorithm provides a tight upper bound in fast CPU times. Our method is compared with the LP-relaxation of (QMKP) and the LP-relaxation of (QPxy)

On the Maximum Affinity Coloring : Complexity in Bipartite Conflict Graphs and Links with Multiway Cut

National audienceThe maximum affinity K-coloring problem is a generalization of the classical K- coloring problem that enables to model that two vertices should be, if possible, colored with the same color. Such vertices are linked with another kind of edges, called affinities. Almost all the algorithms used in practice use local criteria to determine whether an affinity can be satisfied (i.e. its endpoints can be colored with the same color) or not. These criteria only rely on the colorability of the graph and do not take care of the global configuration of affinities.This paper has two objectives. First, highlight a problem that models many applications and on which much work has to be done. Second, point out that affinities have to be considered more globally than they currently are in the literature.We first describe complexity results, mostly in 2-colorable graphs. In partic- ular, we show that the problem is NP-hard in these graphs for K = 2, meaning that the problem is hard even if coloring the graph is easy. Then we prove that any affinity of a graph can be satisfied while preserving the K-colorability in partial (K ? 1)-trees. Finally, we prove that, in partial (K ? 1)-trees, an optimal affinity coloring can be found by solving a minimum multiway cut instance and a classical K-coloring instance separately

An asymptotic linearization for non separable convex and integer quadratic programming

We present an exact method for solving non separable convex integer quadratic problems (IQP). Such problems arise in financial applications. The method we propose transforms (IQP) into a parameterized mixed linear integer problem which provides an overestimation of (IQP) depending on an integer parameter K. We show that as K gets larger, the overestimation tends to the optimal value of (IQP). The practical value of this approach is supported by numerical experiments. The asymptotic behavior of the method, associated with the determination of a precise feasible solution, allows us to exactly solve instances involving up to 60 bounded integer variables. We compare our computational results with the ones obtained by using a commercial solver (Cplex)

Using a Mixed Integer Programming Tool for Solving the 0-1 Quadratic Knapsack Problem

(présenté à ROADEF 2000, Nantes, 26-28 janvier 2000)Abstract. We consider in this paper the 0-1 Quadratic Knapsack Problem (QKP). Our purpose is to show that using a linear reformulation of this problem and a standard mixed integer programming tool, it is possible to efficiently solve (QKP) in terms of computation time and of size of problems, in comparison with the existing methods. Considering a problem involving n variables, the linearization technique we propose has the advantage of adding only (n-1) real variables and 2(n-1) constraints. This method allows to exactly solve (QKP) instances up to 140 variables

Comparaison expérimentale de différentes bornes inférieures pour un problème de placement de tâches

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