Singleton algorithms for the Constraint Satisfaction Problem

A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a constraint to some tuple from the constraint relation, then run the algorithm, and remove the tuple from the constraint if the answer is negative. We characterize the power of the singleton versions of standard universal algorithms for the (promise) CSP over a fixed template in terms of the existence of a minion homomorphism. Using the Hales-Jewett theorem, we show that for finite relational structures this minion condition is equivalent to the existence of polymorphisms with certain symmetries, called palette block symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all tractable CSPs over domains of size at most 7. Finally, by providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of the palette block symmetric polymorphisms.