In this paper we describe and compare two frameworks for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. One is based on a semiring, and the other one on a totally ordered commutative monoid. While comparing the two approaches, we show how to pass from one to the other one, and we discuss when this is possible. The two frameworks have been independently introduced in [2], [3] and [35].

Introduction Constraint satisfaction problems (CSPs) involve finding an assignment of values to variables that satisfy a set of constraints between these variables. In many important applications the problems may be overconstrained, so that no complete solution is possible. In these cases, partial solutions may still be useful if a sufficient number of the most important constraints are satisfied. An example of such partial constraint satisfaction is the maximal constraint satisfaction problem (MAX-CSP), in which the goal is to find assignments of values to variables that satisfy the maximum number of constraints. An important type of CSP is the satisfiability problem (SAT), in which the goal is to find a truth assignment that satisfies a CNF formula in propositional logic. A generalization of satisfiability that corresponds to MAX-CSP is the maximum satisfiability problem (MAX-SAT) in which the object is to find an truth assignment that satisfies the maximum number of clauses

. Problems that do not have complete solutions occur in many areas of application of constraint solving. Heuristic repair methods that have been used successfully on complete CSPs can also be used on overconstrained problems. A difficulty in analyzing their performance is the uncertainty about the goodness of solutions returned in relation to the optimal (best possible) solutions. This difficulty can be overcome by testing these procedures on problems that can be solved by complete methods, which return certifiably optimal solutions. With this experimental strategy, comparative analyses of hill-climbing methods were carried out using anytime curves that could be compared with known optima. In addition, extensive analysis of parameter values for key strategies such as random walk and restarting could be done precisely and efficiently by allowing local search to run until a solution was discovered that was known to be optimal, based on earlier tests with complete methods. An important fi...

Two methods are described for enhancing performance of branch and bound methods for overconstrained CSPs. These methods improve either the upper or lower bound, respectively, during search, so the two can be combined. Upper bounds are improved by using heuristic repair methods before search to find a good solution quickly, whose cost is used as the initial upper bound. The method for improving lower bounds is an extension of directed arc consistency preprocessing, used in conjunction with forward checking. After computing directed arc consistency counts, inferred counts are computed for all values based on minimum counts for values of adjacent variables that are later in the search order. This inference process can be iterated, so that counts are cascaded from the end to the beginning of the search order, to augment the initial counts. Improvements in time and effort are demonstrated for both techniques using random problems. Introduction Constraint satisfaction problems (CSPs) invol...

In this paper we present a new framework for over constrained problems. We suggest to define an over-constrained network as a global constraint. We introduce two new lower bounds of the number of violations, without making any assumption on the arity of constraints.

The constraint satisfaction problem (CSP) is a potential area of application for anytime methods. In this work we derive anytime curves using a partial constraint satisfaction framework that encompasses problems with complete solutions and those that allow only partial solutions of varying quality. In either case, the curves should converge on optimal solutions with respect to some measure of cost (here, the number of violated constraints). Binary CSPs and k-satisfiability problems were tested, using heuristic repair and branch and bound methods. Curves for heuristic methods either start at a lower level than curves for branch and bound (minconflicts with binary CSPs) or have a steeper initial descent (GSAT with k-SAT problems). Techniques for randomization such as random walks or restarting with a new random solution appear to be necessary with heuristic procedures for complete convergence to an optimal solution. Branch and bound algorithms are usefully employed in tandem with heurist...

In recent years, many works have been carried out to solve over-constrained problems, and more specically the Maximal Constraint Satisfaction Problem (Max-CSP), where the goal is to minimize the number of constraint violations. Some lower bounds on this number of violations have been proposed in the literature. In this paper, we characterize the constraints that are ignored by the existing results, we propose new lower bounds which take into account some of these ignored constraints and we show how these new bounds can be integrated into existing ones in order to improve the previous results. Our work also generalize the previous studies by dealing with any kind of constraints, as non binary constraints, or constraints with specic ltering algorithms. Furthermore, in order to integrate these algorithms into any constraint solver, we suggest to represent a Max-CSP as a single global constraint. This constraint can be itself included into any set of constraint. In this way, an over-constrained part of a problem can be isolated from constraints that must be necessarily satised. 1

Abstract. Partial constraint satisfaction [5] was widely studied in the 90’s, and notably Max-CSP solving algorithms [21, 20, 1, 10]. These algorithms compute a lower bound of violated constraints without using propagation. Therefore, recent methods focus on the exploitation of propagation mechanisms to improve the solving process. Soft arc-consistency algorithms [11, 18, 19] propagate inconsistency counters through domains. Another technique consists of using constraint propagation to identify conflict-sets which are pairwise disjoint [16]; the number of conflict-sets extracted leads to a lower bound. In this paper, we place this technique in a more general context. We show this technique reduces to a polynomial case of the NP-Complete Hitting Set problem. Conflict-sets are chosen disjoint to compute the lower bound polynomially. We present a new polynomial case where the conflict-sets share some constraints, and a third case which is not polynomial but such that the cardinality of the Hitting Set can be reasonably under estimated. For each one we provide the algorithm and a schema to generate incrementally the conflict-set collection. We show its extension to weighted CSPs. 1

. Maximum satisfiability (MAX-SAT) is an extension of satisfiability (SAT), in which a partial solution is sought that satisfies the maximum number of clauses in a logical formula. In recent years there has been an growing interest in this and other types of over-constrained problems. Branch and bound extensions of the Davis-Putnam algorithm can return guaranteed optimal solutions to these problems. Earlier work did not make use of a pure literal rule because it appeared to be inefficient here, as for traditional SAT. However, arguments can be adduced to show that pure literals are likely to appear during search for MAX-2SAT, so that fixation of their variables may be effective here. The present work confirms this and also shows that a value ordering heuristic involving literals that are monotone in unit open clauses can be very effective, operating somewhat independently of the ordinary fixation of fully monotone literals. Alone or together, these pure literal strategies can produce i...

The partial constraint satisfaction paradigm focuses on solving relaxations of problems that either do not admit solutions, or that are either impractical or impossible to solve completely. The semiring-based framework for soft constraints is a unifying model for a variety of extensions of the constraint satisfaction formalism. For example, the semiring-based framework can represent weighted, fuzzy, probabilistic and set-based constraint satisfaction problems. In this paper, we discuss how the semiring-based framework for soft constraints can be used to model partial constraint satisfaction problems. We show how the semiring framework can be used to capture a notion of distance between a solution and a problem based on the known distance metrics used in the partial constraint satisfaction literature. These solution-problem distance metrics can be seen as providing lower-bounds on the distance between a problem and its relaxation. 1.