# What is local consistency in constraint satisfaction problems?

In the Constraint Propagation in CSP, it is often stated that pre-processing can solve the whole problem, so no search is required at all. And the key idea is local consistency. What does this actually mean?

• Have a look at https://en.wikipedia.org/wiki/Local_consistency.
– nbro
Commented May 2, 2019 at 13:03
• I think the consistency propagation method only works for 2-SAT problems, no? Commented Sep 6, 2022 at 5:28

If we can do some reduction in the search space using CSP (constraint propagation) we can drastically reduce the search space or sometimes completely avoid the need for a search by directly reaching the solutions (for e.g. with variables having their domains reduced to size one). It could also happen that we come to a point when a variable domain size becomes zero, in that case no solution exists, given the constraints, so no need for a search.

Constraint propagation basically involves the concept of enforcing local consistency (this is done by enforcing node-consistency, arc-consistency, path-consistency and also global constraints using Alldiif or Atmost).

The terms: nodes, arc, path, etc. basically reflects a CSP problem represented as a graph with nodes as the variables and the arcs/edges as constraints. The process is simply to remove values from the domains of the variables that are inconsistent. Algorithms such as AC-3, PC-2, etc. precisely are for these purposes.