The Element constraint

The subtour-elimination filtering

The Sum constraint

The Alldifferent constraint

0 -> 4

0 -> 12

4 -> 0

6 -> 8

12 -> 0

8 -> 6

Achieving domain consistency for a Circuit constraint is an NP-hard problem.

The consistency test that verifies that there is a unique strongly connected component can be achieved in O(n) time, where n is the number of variables in the scope of the Circuit constraint.

The subtour-elimination filtering algorithm runs in O(n) time in the worst case, which can be lowered to O(number of new fixed variables).

LNS is a local-search approach where CP can be used to explore the neighborhood.

A restart in LNS is always caused by reaching the limit on the number of failures.

Increasing the failure limit and the number of relaxed variables augments the time between two restarts.

LNS with a relaxation of 5% of randomly selected variables can always prove the optimality if enough restarts are performed.

Increasing the number of relaxed variables increases the time between two restarts.

A restart in LNS can be caused because the restricted problem has been exhaustively explored.

The first solution with LNS needs to be found by CP.