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Incoming links: Minimum remaining values
Degree heuristic 2026-10-25
Choose an unassigned variable constrained to the greatest number of other unassigned variables. This is commonly a tie-breaker after minimum remaining values. The relevant vertex degree is computed in the remaining constraint graph, rather than counting neighbors already assigned. It aims to make each decision constrain many future decisions without changing the solution set.
Minimum remaining values Created 2026-10-20 Updated 2026-11-05
The minimum remaining values heuristic chooses an unassigned variable with the smallest current legal domain during constraint search. It tries the most constrained choice first: a domain of size zero proves failure, and size one forces a value. If one variable has two remaining digits and another has seven, the first is selected, with a separate rule for ties. This can expose contradictions earlier but does not guarantee a minimum search tree. The heuristic depends on domains after the chosen propagation method, not simply on each variable's original domain size.
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.
The translation gives a sound and complete route through an optimized SAT solver; it trades access to general-purpose SAT machinery against encoding cost and the possible loss of constraint-specific structure. Modern SAT solvers use efficient unit propagation, adaptive branching and clause learning. Learned Boolean clauses can summarize conflicts involving many original variables, allowing later branches to avoid repeating the same failed combination. The reduction also offers a standard, reusable interface: improvements to a SAT solver can benefit many finite-domain constraint satisfaction problems.
The Boolean formula may be much larger than the original constraint satisfaction problem, especially for a succinct high-arity relation. One original variable becomes many indicators, and general branching may lose the interpretation that the minimum remaining values and degree heuristic exploit directly. Native constraint propagation can exploit a relation without enumerating all its forbidden tuples. Even in the binary case, stronger propagation does not follow just from switching to SAT.
For example, take , , and let the only allowed pairs be . The forbidden-pair encoding contains and . If becomes false, arc consistency deletes because it has no support. Yet the direct encoding has no unit Boolean clause forcing : both remain possible individually, so plain unit propagation does not make that deletion. A SAT solver can eventually derive it through additional reasoning or search. Conversely, clause learning can combine information beyond the local consistency enforced by AC-3. Neither approach uniformly dominates: performance depends on representation, propagation strength and the instance's structure; both still face exponential worst-case search on general inputs.