Propagation des contraintes tables souples Etude pr eliminaire

National audienceWCSP is a framework that has attracted a lot of at- tention during the last decade. In particular, there have been many developments of ltering approaches based on the concept of soft local consistencies such as node consistency (NC), arc consistency (AC), full directio- nal arc consistency (FDAC), existential directional arc consistency (EDAC), virtual arc consistency (VAC) and optimal soft arc consistency (OSAC). Almost all algo- rithms related to these properties have been introduced for binary weighted constraint networks, and most of the conducted experiments typically include constraint networks involving only binary and ternary constraints. In this paper, we focus on extensional soft constraints of large arity. We propose an algorithm to lter such constraints and embed it in PFC-MRDAC.Durant ces dix derni ères ann ées, de nombreuses études ont ét és r éalis ées pour le cadre WCSP (Weighted Constraint Satisfaction Problem). En particulier, ont ét é propos ées de nombreuses techniques de filtrage bas ées sur le concept de coh érence locale souple telle que la co- h érence de n oeud, et surtout la coh érence d'arc souple. Toutefois, la plupart de ces algorithmes ont ét és intro- duits pour le cas des contraintes binaires, et la plupart des exp érimentations ont ét és men ées sur des r éseaux de contraintes comportant uniquement des contraintes binaires et/ou ternaires. Dans cet article, nous nous in- t eressons aux contraintes tables souples de grande arit é. Nous proposons un premier algorithme pour filtrer ces contraintes et nous l'int égrons a PFC-MRDAC

Extension des cohérences WCSP aux tuples

National audienceDans cet article, nous présentons un nouveau type de propriétés pour les réseaux de contraintes pondérées (WCNs pour Weighted Constraint networks). Il s'agit de la cohérence de tuples (TC) dont l'établissement sur un WCN est effectué grâce à une nouvelle opération appelée TupleProject. Nous proposons également une version "optimale" de cette propriété, OTC, qui peut être perçue comme une généralisation de OSAC (Optimal Soft Arc Consistency). Le principe sous-jacent à OTC est d'appliquer de manière itérative l'opération TupleProject afin de factoriser un coût qui maximise la borne inférieure w;, sur la base de transferts de coûts entre tuples de différentes contraintes d'arité quelconque

The Power of Linear Programming for Valued CSPs

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo

On Global Warming (Softening Global Constraints)

We describe soft versions of the global cardinality constraint and the regular constraint, with efficient filtering algorithms maintaining domain consistency. For both constraints, the softening is achieved by augmenting the underlying graph. The softened constraints can be used to extend the meta-constraint framework for over-constrained problems proposed by Petit, Regin and Bessiere.Comment: 15 pages, 7 figures. Accepted at the 6th International Workshop on Preferences and Soft Constraint

Maximum Persistency via Iterative Relaxed Inference with Graphical Models

We consider the NP-hard problem of MAP-inference for undirected discrete graphical models. We propose a polynomial time and practically efficient algorithm for finding a part of its optimal solution. Specifically, our algorithm marks some labels of the considered graphical model either as (i) optimal, meaning that they belong to all optimal solutions of the inference problem; (ii) non-optimal if they provably do not belong to any solution. With access to an exact solver of a linear programming relaxation to the MAP-inference problem, our algorithm marks the maximal possible (in a specified sense) number of labels. We also present a version of the algorithm, which has access to a suboptimal dual solver only and still can ensure the (non-)optimality for the marked labels, although the overall number of the marked labels may decrease. We propose an efficient implementation, which runs in time comparable to a single run of a suboptimal dual solver. Our method is well-scalable and shows state-of-the-art results on computational benchmarks from machine learning and computer vision.Comment: Reworked version, submitted to PAM

A Logical Approach to Efficient Max-SAT solving

Weighted Max-SAT is the optimization version of SAT and many important problems can be naturally encoded as such. Solving weighted Max-SAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in finding efficient solving techniques. Most of this work focus on the computation of good quality lower bounds to be used within a branch and bound DPLL-like algorithm. Most often, these lower bounds are described in a procedural way. Because of that, it is difficult to realize the {\em logic} that is behind. In this paper we introduce an original framework for Max-SAT that stresses the parallelism with classical SAT. Then, we extend the two basic SAT solving techniques: {\em search} and {\em inference}. We show that many algorithmic {\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in {\em logic} terms with our framework in a unified manner. Besides, we introduce an original search algorithm that performs a restricted amount of {\em weighted resolution} at each visited node. We empirically compare our algorithm with a variety of solving alternatives on several benchmarks. Our experiments, which constitute to the best of our knowledge the most comprehensive Max-sat evaluation ever reported, show that our algorithm is generally orders of magnitude faster than any competitor

Bornes inférieures à base d'inégalités valides pour les WCSP

La plus part des algorithmes de résolution efficace de WCSP se basent sur la notion de consistance d'arc utilisée pour transformer un WCSP en un WCSP équivalent et plus facile à résoudre. Dans ce but, plusieurs formes de consistance d'arc ont été proposées : AC* \cite{Schiex.00}, DAC* \cite{Larrosa.03}, FDAC* \cite{Larrosa.03},EDAC* \cite{deGivry.05}. Récemment, une consistance d'arc optimale (OSAC pour Optimal Soft Arc Consistency) \cite{Cooper.07} a été proposée. Elle se base sur la résolution d'un Programme Linéaire. Son inconvénient réside dans le fait qu'elle nécessite beaucoup de temps de calcul. Cet inconvénient est dû à la taille du programme linéaire résolu. Nous proposons une nouvelle technique de transformation d'un WCSP en un WCSP équivalent. Cette technique se base sur la modélisation du WCSP sous forme d'un programme linéaire plus facile à résoudre que le calcul de OSAC

Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms

Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization