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Incoming links: Forward checking
CS 188 constraint filtering notes 2026-10-24
Official course-hosted technical exposition: 2.3 Filtering.
Relevant locator: §2.3, initial forward-checking definition and subsequent directed support-revision procedure. These describe forward checking, arc consistency and iterative filtering.
Forward checking Created 2026-10-20 Updated 2026-11-05
Forward checking in a constraint satisfaction problem updates the domains of unassigned variables after assigning one variable. Remove any remaining value immediately incompatible with the new assignment; if a domain becomes empty, backtrack. For with both domains , assigning removes one from the domain of . Basic forward checking only enforces constraints involving the latest assigned value; it does not generally propagate every interaction among still-unassigned variables. Stronger constraint propagation can therefore detect failures earlier at additional computational cost.
Backtracking assigns one variable at a time and abandons a partial assignment as soon as it cannot extend to a solution. With forward checking, placing a knight deletes its occupied and attacked squares from every remaining knight's constraint domain. An empty constraint domain rejects the branch immediately; stronger constraint propagation, such as arc consistency, can detect additional incompatibilities among future choices.
This combines early rejection with depth-first storage. It retains the current assignment and domain changes rather than the entire breadth-first layer of partial placements. A direct knight-variable implementation uses space for explicit domains, plus its assignment and restoration trail, instead of an exponentially large breadth-first search search frontier. Propagation has a computational cost and does not guarantee improvement on every input, but often prevents exploration of vast inconsistent subtrees.
The minimum remaining values heuristic selects an unassigned variable with the fewest currently legal values, exposing the likeliest failure first. The degree heuristic selects a variable constrained to the largest number of other unassigned variables, often breaking minimum remaining values heuristic ties. Degree is measured in the remaining constraint graph, not by counting assigned neighbors. In the symmetric knight-variable encoding the remaining degrees can all tie, so the heuristic need not discriminate there.[ref]
CS 188 constraint ordering and structure notes. §2.4, Minimum Remaining Values: choose an unassigned variable with the fewest valid remaining values
For a finite constraint satisfaction problem, remove values violating unary restrictions before starting. The following backtracking algorithm makes independent copies of constraint domains at each branch, so failed forward checking cannot contaminate a sibling branch.
SOLVE(P):
D := domains of P after unary filtering
if some D[X] is empty: return failure
return SEARCH({}, D)
SEARCH(assignment, D):
if every variable is assigned: return assignment
X := SELECT-UNASSIGNED-VARIABLE(assignment, D, P)
for v in ORDER-DOMAIN-VALUES(X, assignment, D, P):
if X=v is compatible with the current assignment:
next_assignment := assignment extended by X=v
next_domains := a deep copy of D
next_domains[X] := {v}
if INFERENCES(X, v, next_assignment, next_domains, P) succeeds:
result := SEARCH(next_assignment, next_domains)
if result is not failure: return result
return failureSELECT-UNASSIGNED-VARIABLE implements minimum remaining values: choose an unassigned variable with the smallest current constraint domain. Break ties by the degree heuristic, choosing the variable constrained to the greatest number of other unassigned variables; a fixed final tie rule makes execution reproducible. ORDER-DOMAIN-VALUES may use any complete ordering; ascending values suffice here.
INFERENCES implements forward checking: for each unassigned neighbor of , remove every value incompatible with , and fail if any constraint domain becomes empty. For higher-arity constraints, reject any completely instantiated violation and use a sound chosen filtering rule; the stated neighbor test is the usual binary case. Forward checking never removes a value used by a possible extension of the current assignment. Exhausting every surviving value therefore makes this finite backtracking procedure both sound and complete.[ref]
CS 188 constraint filtering notes. §2.3, opening paragraphs: filtering neighbors after assignment
[ref]Forward checking makes minimum remaining values respond to current restrictions rather than the original constraint domain sizes. After , its neighbors may have fewer choices, become forced or lose all choices. Minimum remaining values then selects the newly constrained variable early, while an empty constraint domain triggers immediate failure. The two mechanisms expose failure sooner and reduce wasted backtracking.
Forward checking only tests an unassigned neighbor against the newly assigned value. It need not check whether two still-unassigned variables have mutually supporting values. Arc consistency requires every value at either end of each binary constraint to have support in the other constraint domain, and AC-3 propagates deletions until this condition stabilizes or a constraint domain becomes empty.[ref]
For example, let initially have constraint domains with constraints and . Assigning makes forward checking reduce to , but leaves . Arc consistency also deletes from , leaving . Neither procedure alone decides every general constraint satisfaction problem: a triangle of two-valued inequalities satisfies arc consistency but is unsatisfiable.
One solution is . Use ascending value order; after minimum remaining values and the degree heuristic, break any remaining variable tie alphabetically. Apply the unary restrictions first:
Minimum remaining values selects , even though has greater initial vertex degree. Set . Forward checking removes from the three neighbors :Now is the unique minimum remaining values choice. Set . Forward checking removes from , givingSelect and set . Its remaining neighbors are , so forward checking givesThe two remaining variables tie on constraint domain size and each has one unassigned neighbor, so the degree heuristic also ties. Alphabetical order selects ; take . Since is already absent from , forward checking makes no further deletion. Finally choose . No empty constraint domain or violation occurs, so this branch needs no backtracking.
All eight binary constraints are checked explicitly by the unequal value pairsThe unary restrictions also hold. Equalities and are permitted because those pairs have no edges in the constraint graph.
A solution is . Here is a backtracking search with forward checking, listing the chosen value order rather than assuming ascending values. No backtrack is needed on this branch.
Start with the constraint domains in (a). Assign ; remove from . Choose ; remove from each remaining neighbour. Now both have domain and each have . Choose . Its inequality removes from , forcing , and also removes from , leaving . Assigning removes from , leaving .
Choose . The edge reduces to ; makes no further deletion because is already absent from . Choose and then . All variables are assigned, and every constrained pair has different values.
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