Third edition of Artificial Intelligence: foundations of computational agents, Cambridge University Press, 2023 is now available (including the full text).

### 4.2.2 Constraint Satisfaction Problems

A **constraint satisfaction problem** (CSP) consists of

- a set of variables,
- a domain for each variable, and
- a set of constraints.

The aim is to choose a value for each variable so that the resulting possible world satisfies the constraints; we want a model of the constraints.

A finite CSP has a finite set of variables and a finite domain for each variable. Many of the methods considered in this chapter only work for finite CSPs, although some are designed for infinite, even continuous, domains.

The multidimensional aspect of these problems, where each variable can be seen as a separate dimension, makes them difficult to solve but also provides structure that can be exploited.

Given a CSP, there are a number of tasks that can be performed:

- Determine whether or not there is a model.
- Find a model.
- Find all of the models or enumerate the models.
- Count the number of models.
- Find the best model, given a measure of how good models are; see Section 4.10.
- Determine whether some statement holds in all models.

This chapter mostly considers the problem of finding a model. Some of the methods can also determine if there is no solution. What may be more surprising is that some of the methods can find a model if one exists, but they cannot tell us that there is no model if none exists.

CSPs are very common, so it is worth trying to find relatively efficient ways to solve them. Determining whether there is a model for a CSP with finite domains is NP-hard (see box) and no known algorithms exist to solve such problems that do not use exponential time in the worst case. However, just because a problem is NP-hard does not mean that all instances are difficult to solve. Many instances have structure that can be exploited.