Reoptimization in lagrangian methods for the quadratic knapsack problem

International audienceThe 0-1 quadratic knapsack problem consists in maximizing a quadratic objective function subject to a linear capacity constraint. To solve exactly large instances of this problem with a tree search algorithm (e.g. a branch and bound method), the knowledge of good lower and upper bounds is crucial for pruning the tree but also for fixing as many variables as possible in a preprocessing phase. The upper bounds used in the best known exact approaches are based on Lagrangian relaxation and decomposition. It appears that the computation of these Lagrangian dual bounds involves the resolution of numerous 0-1 linear knapsack subproblems. Thus, taking this huge number of solvings into account, we propose to embed reoptimization techniques for improving the efficiency of the preprocessing phase of the 0-1 quadratic knapsack resolution. Namely, reoptimization is introduced to accelerate each independent sequence of 0-1 linear knapsack problems induced by the Lagrangian relaxation as well as the Lagrangian decomposition. Numerous numerical experiments validate the relevance of our approach

A greedy algorithm for multicut and integral multiflow in rooted trees

We present an O(min(Kn,n2)) algorithm to solve the maximum integral multiflow and minimum multicut problems in rooted trees, where K is the number of commodities and n is the number of vertices. These problems are NP-hard in undirected trees but polynomial in directed trees. In the algorithm we propose, we first use a greedy procedure to build the multiflow then we use duality properties to obtain the multicut and prove the optimality

Reduced graphs for min-cut/max-flow approaches in image segmentation

International audienceIn few years, min-cut/max-flow approach has become a leading method for solving a wide range of problems in computer vision. However, min-cut/max-flow approaches involve the construction of huge graphs which sometimes do not fit in memory. Currently, most of the max-flow algorithms are impracticable to solve such large scale problems. In this paper, we introduce a new strategy for reducing exactly graphs in the image segmentation context. During the creation of the graph, we test if the node is really useful to the max-flow computation. Numerical experiments validate the relevance of this technique to segment large scale images

Extraction de biclusters contraints dans des contextes bruités

National audienceL'extraction de biclusters, qui consiste à rechercher un groupe d'attributs qui montrent un comportement cohérent pour un sous-ensemble d'observations dans une matrice de données, est une tâche importante dans divers domaines, telle que la biologie. Nous proposons ici un nouveau système, COBIC, qui combine des algorithmes de graphes avec des méthodes de fouille de données pour une recherche efficace de biclusters pertinents et susceptibles de se recouvrir. COBIC est fondé sur les algorithmes de flot maximal/coupe minimale et est capable de prendre en compte les connaissances d'une base exprimées sous forme d'une classification, par un mécanisme d'adaptation des poids lors de l'extraction itérative des régions denses. L'évaluation de COBIC sur des données réelles et la comparaison par rapport à des méthodes efficaces de biclustering montrent que COBIC est très performant et en particulier lorsque la qualité des biclusters s'évalue en fonction de la significativité de l'enrichissement des clusters calculés avec les fonctions cellulaires décrites dans l'Ontologie GO

Multicut and integral multiflow : a survey.

We present a survey about the maximum integral multiflow and minimum multicut problems and their subproblems, such as the multiterminal cut and the unsplittable flow problems. We consider neither continuous multiflow nor minimum cost multiflow. Most of the results are very recent and some are new. We recall the dual relationship between both problems, give complexity results and algorithms, firstly in unrestricted graphs and secondly in several special graphs: trees, bipartite or planar graphs. A table summarizes the most important results

Dantzig-Wolfe and Lagrangian decompositions in integer linear programming

International audienceWe propose in this paper a new Dantzig-Wolfe master model based on Lagrangian decomposition. We establish the relationship with classical Dantzig-Wolfe decomposition master problem and propose an alternative proof of the dominance of Lagrangian decomposition on Lagrangian relaxation dual bound. As illustration, we give the corresponding models and numerical results for two standard mathematical programs: the 0-1 bidimensional knapsack problem and the generalized assignment problem

An improving dynamic programming algorithm to solve the shortest path problem with time windows

International audienceAn efficient use of dynamic programming requires a substantial reduction of the number of labels. We propose in this paper an efficient way of reducing the number of labels saved and dominance computing time. Our approach is validated by experiments on shortest path problem with time windows instances

Solutions diversification in a column generation algorithm

International audienceColumn generation algorithms have been specially designed for solving mathematical programs with a huge number of variables. Unfortunately, this method suffers from slow convergence that limits its efficiency and usability. Several accelerating approaches are proposed in the literature such as stabilization-based techniques. A more classical approach, known as "intensification, consists in inserting a set of columns instead of only the best one. Unfortunately, this intensication typically overloads the master problem, and generates a huge number of useless variables. This article covers some characteristics of the generated columns from theoretical and experimental points of view. Two selection criteria are compared. The first one is based on column reduced cost and the second on column structure. We conclude our study with computational experiments on two kinds of problems: the acyclic vehicle routing problem with time windows and the one-dimensional cutting stock. problem

Une méthode de réduction exacte pour la segmentation par graph cuts

8 pagesLes graph cuts sont désormais un standard au sein de la communauté de la vision par ordinateur. Néanmoins, leur grande consommation mémoire reste un problème majeur : les graphes sous-jacents contiennent des milliards de noeuds et davantage d'arcs. Excepté quelques méthodes [14, 10, 5] exactes, les heuristiques présentes dans la littérature ne permettent d'obtenir qu'une solution approchée [12, 8]. Dans un premier temps, nous présentons une nouvelle stratégie pour réduire exactement ces graphes : le graphe est construit en ajoutant les noeuds qui satisfont localement une condition donnée et correspond à une bande étroite autour des contours de l'objet à segmenter. Les expériences présentées pour segmenter des images en niveaux de gris et en couleur mettent en évidence une faible consommation mémoire tout en garantissant une faible distance sur les segmentations. Nous présentons aussi une application de cette méthode pour segmenter des tumeurs dans des images scanner