Singleton algorithms for the Constraint Satisfaction Problem
This work addresses theoretical computer scientists by providing a deeper understanding of algorithm variants for CSPs, though it is incremental as it builds on existing universal algorithms and minion theory.
The paper tackles the problem of characterizing the power of singleton versions of algorithms for constraint satisfaction problems (CSPs) by linking them to minion homomorphisms and palette symmetric polymorphisms, showing that singleton BLP+AIP solves all tractable CSPs over domains up to size 7 but fails for domain size 8 due to a structure from the dihedral group D4.
A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a constraint to a 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, 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 $D_4$ 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. Surprisingly, we also identify a concrete temporal relational structure whose CSP can be solved by a singleton algorithm after sampling, yet it does not admit palette symmetric polymorphisms. This highlights that the Hales-Jewett argument applies only to finite structures. Nevertheless, we introduce generalized palette polymorphisms for every tractable temporal relational structure and show that they yield a rounding procedure implying that the corresponding singleton algorithm solves its CSP after sampling.