Constraint Satisfaction Problems Questions and Answers

Answer: C

If relabelling colours or reflecting a board produces another solution, the search wastes effort exploring every copy, and adding an ordering constraint keeps only one representative of each class. Care is needed not to exclude all solutions. Constraint removal, forced equality and ordering for preservation are different operations. Recognising that a search is exploring the same thing many times over, and forbidding the duplicates, can reduce the work by an enormous factor.

Answer: D

A constraint intended to keep one representative must be shown to leave at least one member of each equivalence class, or the problem becomes unsatisfiable although solutions exist. That is easy to get wrong when several symmetries interact. Propagation usually improves, the variable count is unchanged, and the constraints are added before searching. A modification justified by an argument must have that argument checked, since the failure is silent and looks like an insoluble problem.

Answer: A

The standard formulation makes each region a variable whose domain is the available colours, with a binary inequality constraint between every pair of adjacent regions. It is the textbook example because the constraint graph is the map's adjacency graph and can be drawn. The alternatives invert the roles or collapse the structure. Choosing what to make a variable is the first modelling decision and it determines everything that follows.

Answer: B

The graph is literally the map's adjacency structure, so the effect of every heuristic and of propagation can be seen directly on a picture. Australia's seven states make a particularly convenient instance. Arc consistency applies far more widely, solutions are usually numerous, and every method applies. A worked example whose structure can be seen at a glance is worth a great deal when the general case must be reasoned about abstractly.

Answer: C

Making each letter a variable over the digits, adding an all different constraint and writing one constraint per column with an auxiliary carry variable is the standard model. The carries are the modelling subtlety, since without them the column constraints cannot be written locally. The alternatives invert the roles or discard the structure. Introducing auxiliary variables to make constraints local is a technique that recurs throughout constraint modelling.

Answer: D

Without carries the addition relates every column at once, but with them each column involves only its own letters and two carries, so propagation works locally and effectively. The variable count rises, which is a price worth paying. All different is independent, and solution count is a property of the puzzle. Adding variables to reduce the arity of constraints is a counter-intuitive but standard modelling move, since propagation strength usually matters more than problem size.

Answer: A

Making each task's start time a variable, with precedence constraints between tasks and resource constraints limiting overlap, is the standard model, and specialised global constraints such as cumulative handle the resources efficiently. The alternatives are possible but propagate far worse. Scheduling is the commercially most important application of the whole framework, which is why the specialised propagators for it are so highly developed.

Answer: B

A cumulative resource constraint reasons about the total demand over a time window and prunes far more than pairwise non-overlap constraints could, which is the difference between solving an industrial instance and not. Search is still required, tasks are fixed by the problem, and the structure remains cyclic. That the strength of a propagator, rather than the search algorithm, decides whether a real instance is solvable is the central practical lesson of constraint programming.

Answer: C

A propositional formula is a constraint problem whose variables are boolean and whose constraints are clauses, so the two literatures address the same territory from different directions and share unit propagation, clause learning and restarts. Neither is strictly a special case of the other in practice, since each has developed its own encodings. Recognising two communities working on the same problem is what allows techniques to be transferred between them.

Answer: D

Learning a constraint from each failure, so the same doomed combination is never reconstructed, transformed satisfiability solving and was carried across as no-good learning. Restarts also transferred and are genuinely useful, though the learning is the larger contribution. Constraint arity and boolean encoding run the other way. Techniques crossing between neighbouring fields, once the correspondence is noticed, is a recurring source of progress.