Constraint Programming

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 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