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 variable to an element from its domain, then run the algorithm, and remove the element from the domain if the answer is negative. Using the Hales-Jewett theorem, 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 polymorphisms with certain symmetries, which we call palette symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all (multi-sorted) tractable CSPs over domains of size at most 7. We further show that already for domain size 8 there exists a relational structure arising from the dihedral group $\mathbf D_4$ that does not admit palette symmetric polymorphisms and cannot be solved by singleton BLP+AIP. By providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of palette symmetric polymorphisms. Among tractable temporal templates, we exhibit a structure demonstrating that finiteness is crucial for the Hales-Jewett argument; nevertheless, by introducing generalized palette polymorphisms we establish tractability for each such template.