Conflict-Directed Backjumping Revisited

TL;DR

This paper investigates the interplay between look-ahead and look-back techniques in backtracking algorithms for constraint satisfaction problems, revealing conditions where conflict-directed backjumping (CBJ) is redundant or beneficial.

Contribution

It provides a theoretical analysis of how variable ordering and local consistency levels affect the usefulness of CBJ, and empirically demonstrates significant speedups with GAC-CBJ.

Findings

Existence of a perfect variable ordering making CBJ redundant.

Higher local consistency levels reduce the benefit of backjumping.

GAC-CBJ can significantly speed up backtracking algorithms maintaining arc consistency.

Abstract

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.