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Joined 2026-09-21 Message
Incoming links: CS 188 constraint ordering and structure notes
Cutset conditioning 2026-10-25
Enumerate assignments to a cycle cutset, reject internal violations, filter all crossing binary constraints, and solve the residual forest by tree-structured constraint satisfaction. A full solution restricts to one enumerated cutset assignment; the residual tree passes preserve all its values. Conversely, a consistent cutset assignment combined with supported residual solutions satisfies every constraint. These observations prove completeness and soundness.
With variables, maximum constraint domain size and cutset size , there are at most assignments and residual tree work per assignment. Internal cutset and crossing-constraint tests require additional accounting. The method is attractive when a small cycle cutset is available.[ref]
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]Cutset conditioning selects a cycle cutset : deleting its vertices makes the binary constraint graph a forest. Enumerate assignments to , reject those violating internal constraints, and filter each remaining constraint domain against the chosen cutset values. Solve the residual components by tree-structured constraint satisfaction. Stop at a successful assignment, or report failure after all cutset assignments fail. Completeness follows by considering the restriction of any full solution to ; soundness follows by checking internal, crossing and residual constraints. With maximum constraint domain size and cutset variables, a typical bound is for the residual forest work, with cutset and crossing checks added explicitly.[ref]
Here take . The residual constraint graph has vertices and only edges , a tree rooted at . Choose the consistent cutset values . Filtering against these assignments givesThe upward support pass from deletes , because can only equal . The pass from retains , which has support or . Assign at the root and then supported child values . Together with , this gives the solution in (c).
No singleton cycle cutset exists: deleting or leaves, respectively, a triangle on or ; deleting leaves the cycle in a graph . Thus the two-element choice is also minimum, although the algorithm only requires a valid cutset.