Maintaining%20Arc%20Consistency

1
Maintaining Arc Consistency
We have a constraint graph G of variables
X1,...Xn, and constraint relations Xi ?Xj,
and each Xi has a value set V(Xi ). An arc from
Xi to Xj is consistent if ?v ? V(Xi ) ?w ? V(Xj)
? v,w is consistent Conversely, an arc is
inconsistent if ? v ? V(Xi ) ? w ? V(Xj) ? v,w
is inconsistent. The arc can be made consistent
by removing v.
Q1
Q2
Q3
Q4
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Arc Consistency (4 queens problem) Example 1
Remove a value v of Qx if there is a variable
Qy such that v is inconsistent with all remaining
values of Qy
- 5 9
- 4 8
- 2 7
1 3 6
4 3 2 1
Numbers indicate sequence of values that are
deleted.
Q1 Q2 Q3 Q4
Starting with Q11, arc consistency check proves
this inconsistent without trying any more
assignments.
3
Arc Consistency (4 queens problem)Example 2
Remove a value v of Qx if there is a variable
Qy such that v is inconsistent with all remaining
values of Qy
- 6 9
- 3 5
2 4 8
- 1 7
4 3 2 1
Numbers indicate sequence of values that are
deleted.
Q1 Q2 Q3 Q4
Starting with Q12, arc consistency check finds a
solution without trying any more assignments.