Conflict-Directed Backjumping Revisited
  In recent years, many improvements to backtracking algorithms for solving
constraint satisfaction problems have been proposed. The techniques for
improving backtracking algorithms can be conveniently classified as look-ahead
schemes and look-back schemes. Unfortunately, look-ahead and look-back schemes
are not entirely orthogonal as it has been observed empirically that the
enhancement of look-ahead techniques is sometimes counterproductive to the
effects of look-back techniques. In this paper, we focus on the relationship
between the two most important look-ahead techniques---using a variable
ordering heuristic and maintaining a level of local consistency during the
backtracking search---and the look-back technique of conflict-directed
backjumping (CBJ). We show that there exists a "perfect" dynamic variable
ordering such that CBJ becomes redundant. We also show theoretically that as
the level of local consistency that is maintained in the backtracking search is
increased, the less that backjumping will be an improvement. Our theoretical
results partially explain why a backtracking algorithm doing more in the
look-ahead phase cannot benefit more from the backjumping look-back scheme.
Finally, we show empirically that adding CBJ to a backtracking algorithm that
maintains generalized arc consistency (GAC), an algorithm that we refer to as
GAC-CBJ, can still provide orders of magnitude speedups. Our empirical results
contrast with Bessiere and Regin's conclusion (1996) that CBJ is useless to an
algorithm that maintains arc consistency.
