# How to deal with Constraints having Large or Infinite Domains.

This presentation gives an overview of Constraint Satisfaction Problems (CSPs). An example they give is this:

• Variables: $X = \{X_1, X_2, X_3\}$
• Domains: $D(X_1) = \{1,2\}, D(X_2) = \{0,1,2,3\}, D(X_3) = \{2,3\}$
• Constraints: $X_1 > X_2 \land X_1 + X_2 = X_3 \land X_1 ≠ X_2 ≠ X_3 ≠ X_1$
• Solution: $X_1 = 2$, $X_2 = 1$, $X_3 = 3$ $\quad alldifferent([X_1, X_2, X_3])$

However the domains are tiny, $D(X_1) = \{1,2\}$, etc. just 2 values. I am wondering how they handle the situation where a variable is say a start time and can take any possible time value (assuming millisecond resolution integer representation of temporal values). So the domain could be $D(X_t) = \{1,\dotsc,10^{10}\}$ or something large. The backtracking algorithm does essentially $\forall d \in D(X_t), \dots$, where you enumerate over every value in the domain. I don't see how this is possible in practice if you are working with time (for example), or any large integer values or more generic data values.

Wondering if one could shed some light on how to deal with Constraint Satisfaction Problems where you have a large or infinite domain, because enumerating all the values seems impractical.