Sandwiches for Promise Constraint Satisfaction

Abstract

Promise Constraint Satisfaction Problems (PCSP) were proposed recently by Brakensiek and Guruswami [2] as a framework to study approximations for Constraint Satisfaction Problems (CSP). Informally a PCSP asks to distinguish between whether a given instance of a CSP has a solution or not even a specified relaxation can be satisfied. All currently known tractable PCSPs can be reduced in a natural way to tractable CSPs. Barto [1] presented an example of a PCSP over Boolean structures for which this reduction requires solving a CSP over an infinite structure. We give a first example of a PCSP over Boolean structures which reduces to a tractable CSP over a structure of size 3 but not smaller. Further we investigate properties of PCSPs that reduce to systems of linear equations or to CSPs over structures with semilattice or majority polymorphism.

Finitely Tractable Promise Constraint Satisfaction Problems

“…Is it so for all symmetric Boolean templates? For general Boolean templates, the answer is "No": [12] presents an example that requires a three-element C. However, it is unclear whether there is an upper bound on the size of C for finitely tractable (Boolean) PCSs, and if there is, how it could be computed. There are also natural concepts beyond finite tractability, still stronger than standard tractability.…”