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

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

Cost Function Networks to Solve Large Computational Protein Design Problems

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A Decomposition Technique for Solving {Max-CSP}

International audienceThe objective of the Maximal Constraint Satisfaction Problem (Max-CSP) is to find an instantiation which minimizes the number of constraint violations in a constraint network. In this paper, inspired from the concept of inferred disjunctive constraints intro- duced by Freuder and Hubbe, we show that it is possible to exploit the arc-inconsistency counts, associated with each value of a net- work, in order to avoid exploring useless portions of the search space. The principle is to reason from the distance between the two best values in the domain of a variable, according to such counts. From this reasoning, we can build a decomposition technique which can be used throughout search in order to decompose the current prob- lem into easier sub-problems. Interestingly, this approach does not depend on the structure of the constraint graph, as it is usually pro- posed. Alternatively, we can dynamically post hard constraints that can be used locally to prune the search space. The practical interest of our approach is illustrated, using this alternative, with an experi- mentation based on a classical branch and bound algorithm, namely PFC-MRDAC

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

The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrised by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors (arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213) to appear in SIAM Journal on Computing (SICOMP

Une approche syntaxique pour le problème de la fusion de réseaux de contraintes qualitatives

National audienceDans cet article, nous nous intéressons au problème de la fusion de réseaux de contraintes qualitatives (RCQ) représentant des croyances ou des préférences locales sur les positions relatives d'entités spatiales ou temporelles. Nous définissons deux classes d'opérateurs de fusion d1 et d2 qui, à un ensemble de RCQ définis sur le même formalisme qualitatif et le même ensemble d'entités, associent un ensemble cohérent de configurations qualitatives représentant une vision globale de ces RCQ. Ces opérateurs sont paramétrés par une distance entre relations du formalisme qualitatif considéré et par des fonctions d'agrégation. Contrairement aux précédents opérateurs proposées pour la fusion de RCQ, nous optons pour une approche syntaxique, où chacune des contraintes des RCQ fournis a une influence sur le résultat de la fusion. Nous étudions les propriétés logiques des opérateurs de fusion définis et montrons leur équivalence sous certaines restrictions. Nous montrons que le résultat fourni par l'opérateur d2 correspond à l'ensemble des solutions optimales d'un RCQ pondéré particulier. Afin de calculer ces solutions, un algorithme basé sur la méthode de fermeture par faible composition étendu au cas des RCQ pondérés est proposé