In constraint programming and SAT solving, backjumping (also known as non-chronological backtracking or intelligent backtracking) is an enhancement for backtracking algorithms which reduces the search space. While backtracking always goes up one level in the search tree is used, but the same considerations apply to a dynamic order of evaluation.

Definition

's value if possible, backtracking again otherwise.

as it would normally do.

is called a safe jump.

Establishing whether a jump is safe is not always feasible, as safe jumps are defined in terms of the set of solutions, which is what the algorithm is trying to find. In practice, backjumping algorithms use the lowest index they can efficiently prove to be a safe jump. Different algorithms use different methods for determining whether a jump is safe. These methods have different costs, but a higher cost of finding a higher safe jump may be traded off a reduced amount of search due to skipping parts of the search tree.

Backjumping at leaf nodes

The simplest condition in which backjumping is possible is when all values of a variable have been proved inconsistent without further branching. In constraint satisfaction, a partial evaluation is consistent if and only if it satisfies all constraints involving the assigned variables, and inconsistent otherwise. It might be the case that a consistent partial solution cannot be extended to a consistent complete solution because some of the unassigned variables may not be assigned without violating other constraints.

is called a leaf dead end is a leaf of the search tree (which correspond to nodes having only leaves as children in the figures of this article.)

The backjumping algorithm by John Gaschnig does a backjump only in leaf dead ends. has been tested and resulted inconsistent without the need of branching over another variable.

, the algorithm checks the consistency of the following evaluations:. Since every variable can usually take more than one value, the maximal index that comes out from the check for each value is a safe jump, and is the point where John Gaschnig's algorithm jumps.

.

Backjumping at internal nodes

The previous algorithm only backjumps when the values of a variable can be shown inconsistent with the current partial solution without further branching. In other words, it allows for a backjump only at leaf nodes in the search tree.

An internal node of the search tree represents an assignment of a variable that is consistent with the previous ones. If no solution extends this assignment, the previous algorithm always backtracks: no backjump is done in this case.

required branching, it is because they are consistent with the current assignment. As a result, searching for a prefix that is inconsistent with these values of the last variable does not succeed.

is the recursive search. In particular, the algorithm "knows" that no solution exists from this point on because it comes back to this node instead of stopping after having found a solution.

This return is due to a number of dead ends, points where the algorithm has proved a partial solution inconsistent. In order to further backjump, the algorithm has to take into account that the impossibility of finding solutions is due to these dead ends. In particular, the safe jumps are indexes of prefixes that still make these dead ends to be inconsistent partial solutions.

.

Simplifications

is collected during the visit of its subtree. Finding a safe jump can be simplified by two considerations. The first is that the algorithm needs a safe jump, but still works with a jump that is not the highest possible safe jump.

, and also irrelevant are their other subtrees.

or from one of its ancestors.

This fact can be exploited by collecting, in each node, a set of previously assigned variables whose evaluation suffices to prove that no solution exists in the subtree rooted at the node. This set is built during the execution of the algorithm. When retracting from a node, this set is removed the variable of the node and collected in the set of the destination of backtracking or backjumping. Since nodes that are skipped from backjumping are never retracted from, their sets are automatically ignored.

Graph-based backjumping

. As a result, the algorithm can backjump to the highest index in this set.

The fact that nodes skipped by backjumping can be ignored when considering a further backjump can be exploited by the following algorithm. When retracting from a leaf node, the set of variables that are in constraint with it is created and "sent back" to its parent, or ancestor in case of backjumping. At every internal node, a set of variables is maintained. Every time a set of variables is received from one of its children or descendants, their variables are added to the maintained set. When further backtracking or backjumping from the node, the variable of the node is removed from this set, and the set is sent to the node that is the destination of backtracking or backjumping. This algorithm works because the set maintained in a node collects all variables that are relevant to prove unsatisfiability in the leaves that are descendants of this node. Since sets of variables are only sent when retracing from nodes, the sets collected at nodes skipped by backjumping are automatically ignored.

Conflict-based backjumping

Conflict-based backjumping ( conflict-directed backjumping) is a more refined algorithm and sometimes able to achieve larger backjumps. It is based on checking not only the common presence of two variables in the same constraint but also on whether the constraint actually caused any inconsistency. In particular, this algorithm collects one of the violated constraints in every leaf. At every node, the highest index of a variable that is in one of the constraints collected at the leaves is a safe jump.

While the violated constraint chosen in each leaf does not affect the safety of the resulting jump, choosing constraints of highest possible indices increases the highness of the jump. For this reason, conflict-based backjumping orders constraints in such a way that constraints over lower indices variables are preferred over constraints on higher index variables.

. In other words, excluding common variables, the constraint that has the all lower indices is preferred.

, the lowest collected index identifies a safe jump.

. In particular, the algorithm collects, in each node, all sets coming from its descendants that have not been skipped by backjumping. When retracting from this node, this set is removed the variable of the node and collected into the destination of backtracking or backjumping.

Conflict-directed backjumping was proposed for Constraint Satisfaction Problems by Patrick Prosser in his seminal 1993 paper.

See also

• Backtracking

• Constraint learning

• Conflict-driven clause learning

• Look-ahead (backtracking)

• GRASP (SAT solver)

Notes and references

Bibliography