Path and k-Consistency
Arc consistency considers pairs. Path and k-consistency ask whether compatible assignments can be extended through additional variables.
A
{1,2}
{1,2}
B
{1,2}
{1,2}
C
{1,2}
{1,2}
A Sudoku-unit example
Imagine three cells in one row, each with domain {1,2}. Every binary inequality arc is consistent: each value in one cell has a different value in another. Yet all three cells cannot receive distinct values from only two digits.
Considering all three variables exposes the inconsistency that pairwise checking misses.
Increasing strength and cost
- Arc consistency: every value extends to each connected neighbor.
- Path consistency: compatible assignments to two variables can be extended through a third.
- k-consistency: consistent assignments to k−1 variables can be extended to a kth.
Tradeoff: Stronger consistency can remove more search, but enforcing it costs more than AC-3.
Exercise: Change C's domain to {1,2,3}. What value is C forced to take under the row's distinctness requirement?