Semiring-Based Soft Constraints

The semiring-based formalism to model soft constraint has been introduced in 1995 by Ugo Montanari and the authors of this paper. The idea was to make constraint programming more flexible and widely applicable. We also wanted to define the extension via a general formalism, so that all its instances could inherit its properties and be easily compared. Since then, much work has been done to study, extend, and apply this formalism. This papers gives a brief summary of some of these research activities. 1 Before soft constraints: a brief introduction to constraint programming Constraint programming [1, 42, 60, 74, 68] is a powerful paradigm for solving combinatorial search problems that draws on a wide range of techniques from artificial intelligence, computer science, databases, programming languages, and operations research. Constraint programming is currently applied with success to many domains, such as scheduling, planning, vehicle routing, configuration, networks, and bioinformatics. The basic idea in constraint programming is that the user states the constraints and a general purpose constraint solver solves them. Constraints are just relations, and a constraint satisfaction problem (CSP) states which relations should hold among the given decision variables. For example, in scheduling activities in a company, the decision variables might be the starting times and the durations of the activities, as well as the resources needed to perform them, and the constraints might be on the availability of the resources and on their use for a limited number of activities at a time. Constraint solvers take a real-world problem, represented in terms of decision variables and constraints, and find an assignment of values to all the variables that satisfies all the constraints. Constraint solvers search the solution space either systematically, as with backtracking or branch and bound algorithms, or use forms of local search which may be incomplete. Systematic methods often interleave search and inference, where inference consists of propagating the information contained in one constraint to the neighboring constraints. Such inference, usually called constraint propagation, may reduce the parts of the search space that need to be visited. Rather than trying to satisfy a set of constraints, sometimes people want to optimize them. This means that there is an objective function that tells us the quality of each solution, and the aim is to find a solution with optimal quality. To solve such problems, techniques such as branch and bound are usually used. The initial ideas underlying the whole constraint programming research area emerged in the ’70s with several pioneering papers on local consistency, among which the 1974 paper by Ugo Montanari [63], where for the first time a form of constraint propagation, called path consistency, was defined and studied in depth. Since then, the field has evolved greatly, and theoretical study has been coupled with application work, that has shown the need for several extensions of the classical constraint formalism. The introduction of semiring-based soft constraints lies within this evolution thread. In the classical notion of constraint programming, constraints are relations. Thus a constraint can either be satisfied or violated. In the early ’90s, some attempts had been made to generalize the notion of constraint to an object with more than just two levels of satisfiability. For example, fuzzy constraints [46, 69] allow for the whole range of satisfiability levels between 0 and 1. Then, the quality of a solution is the minimum level of satisfiability of the constraints for that solution. The aim is then to find a solution whose quality is highest. Because fuzzy constraints suffer from the so-called ”drowning effect” (where the worst level of satisfiability ”drowns” all the others), lexicographic constraints were introduced [49], to obtain a more discriminating ordering of the solutions, where also solutions with the same worst level can be distinguished. Another extension to classical constraints are the so-called probabilistic constraints [48], where, in the context of an uncertain model of the real world, each constraint is associated to the probability of being present in the real problem. Solutions are then associated to their conjoint probability (assuming independence of the constraints), and the aim is to find a solution with the highest probability. In weighted constraints, instead, each constraint is given a weight, and the aim is to find a solution for which the sum of the weights of the satisfied constraints is maximal. A very useful instance of weighted constraints are MaxCSPs, where weights are just 0 or 1 (0 if the constraint is violated and 1 if it is satisfied). In this case, we therefore want to satisfy as many constraints as possible. While fuzzy, lexicographic, and probabilistic constraints were defined for modeling purposes, that is, to model real-life situations that could not be faithfully modeled via classical constraints, weighted constraints and MaxCSPs were mainly addressing over-constrained problems, where there are so many constraints that the problem has no solution. In fact, the aim is to satisfy as many constraints as possible, possibly using priorities (modeled by the weights) to have more discriminating power. This second line of reasoning lead also to the definition of the first general framework to extend classical constraints, called partial constraint satisfaction [51]. In partial CSPs, over-constrained problems are addressed by defining a metric over constraint problems, and by trying to find a solution of a problem which is as close as possible to the given one according to the chosen metric. MaxCSPs are then just an instance of partial CSPs, where the metric is based on the number of satisfied constraints. 2 Semiring-based soft constraints: main idea and