### Chapter 4 Constraint Programming [chapter]

Francesca Rossi, Peter van Beek, Toby Walsh
2008 Foundations of Artificial Intelligence
Constraint programming is a powerful paradigm for solving combinatorial search problems that draws on a wide range of techniques from artificial intelligence, operations research, algorithms, graph theory and elsewhere. The basic idea in constraint programming is that the user states the constraints and a general purpose constraint solver is used to solve them. Constraints are just relations, and a constraint satisfaction problem (CSP) states which relations should hold among the given decision
more » ... variables. More formally, a constraint satisfaction problem consists of a set of variables, each with some domain of values, and a set of relations on subsets of these variables. For example, in scheduling exams at an university, the decision variables might be the times and locations of the different exams, and the constraints might be on the capacity of each examination room (e.g., we cannot schedule more students to sit exams in a given room at any one time than the room's capacity) and on the exams scheduled at the same time (e.g., we cannot schedule two exams at the same time if they share students in common). Constraint solvers take a real-world problem like this represented in terms of decision variables and constraints, and find an assignment to all the variables that satisfies the constraints. Extensions of this framework may involve, for example, finding optimal solutions according to one or more optimization criterion (e.g., minimizing the number of days over which exams need to be scheduled), finding all solutions, replacing (some or all) constraints with preferences, and considering a distributed setting where constraints are distributed among several agents. Constraint solvers search the solution space systematically, as with backtracking or branch and bound algorithms, or use forms of local search which may be incomplete. Systematic method often interleave search (see Section 4.3) and inference, where inference consists of propagating the information contained in one constraint to the neighboring constraints (see Section 4.2). Such inference reduces the parts of the search space that need to be visited. Special propagation procedures can be devised to suit specific constraints (called global constraints), which occur often in real life. Such global constraints are an important component in the success of constraint pro-182 4. Constraint Programming gramming. They provide common patterns to help users model real-world problems. They also help make search for a solution more efficient and more effective. While constraint problems are in general NP-complete, there are important classes which can be solved polynomially (see Section 4.4). They are identified by the connectivity structure among the variables sharing constraints, or by the language to define the constraints. For example, constraint problems where the connectivity graph has the form of a tree are polynomial to solve. While defining a set of constraints may seem a simple way to model a real-world problem, finding a good model that works well with a chosen solver is not easy. A poorly chosen model may be very hard to solve. Moreover, solvers can be designed to take advantage of the features of the model such as symmetry to save time in finding a solution (see Section 4.5). Another problem with modeling real-world problems is that many are over-constrained. We may need to specify preferences rather than constraints. Soft constraints (see Section 4.6) provide a formalism to do this, as well as techniques to find an optimal solution according to the specified preferences. Many of the constraint solving methods like constraint propagation can be adapted to be used with soft constraints. A constraint solver can be implemented in any language. However, there are languages especially designed to represent constraint relations and the chosen search strategy. These languages are logic-based, imperative, object-oriented, or rule-based. Languages based on logic programming (see Section 4.7) are well suited for a tight integration between the language and constraints since they are based on similar notions: relations and (backtracking) search. Constraint solvers can also be extended to deal with relations over more than just finite (or enumerated) domains. For example, relations over the reals are useful to model many real-world problems (see Section 4.8). Another extension is to multiagent systems. We may have several agents, each of which has their own constraints. Since agents may want to keep their knowledge private, or their knowledge is so large and dynamic that it does not make sense to collect it in a centralized site, distributed constraint programming has been developed (see Section 4.9). This chapter necessarily covers some of the issues that are central to constraint programming somewhat superficially. A deeper treatment of these and many other issues can be found in the various books on constraint programming that have been written [5, 35, 53, 98, 70,    . Constraint Propagation One of the most important concepts in the theory and practice of constraint programming is that of local consistency. A local inconsistency is an instantiation of some of the variables that satisfies the relevant constraints but cannot be extended to one or more additional variables and so cannot be part of any solution. If we are using a backtracking search to find a solution, such an inconsistency can be the reason for many deadends in the search and cause much futile search effort. This insight has led to: (a) the definition of conditions that characterize the level of local consistency of a CSP (e.g., [49, 95, 104] ), (b) the development of constraint propagation algorithms-algorithms which enforce these levels of local consistency by removing inconsistencies from a CSP (e.g., [95, 104] ), and (c) effective backtracking algorithms F. Rossi, P. van Beek, T. Walsh 183 for finding solutions to CSPs that maintain a level of local consistency during the search (e.g., [30, 54, 68] ). In this section, we survey definitions of local consistency and constraint propagation algorithms. Backtracking algorithms integrated with constraint propagation are the topic of a subsequent section.