Finitely Tractable Promise Constraint Satisfaction Problems

Abstract

The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems. Barto [LICS '19] has shown that a specific PCSP, the problem to find a valid Not-All-Equal solution to a 1-in-3-SAT instance, is not finitely tractable in that it can be solved by a trivial reduction to a tractable CSP, but such a CSP is necessarily over an infinite domain (unless P=NP). We initiate a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractability within a class of templates -the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].

The local-global property for G-invariant terms

“…An interesting class of conditions within the scope of Lemma 8 are the 'doubly cyclic' identities given by the wreath-product Z p ≀ Z p . These conditions recently found an application in the study of finitary tractable PCSPs [AB20]. We remark that Lemma 8 does not exclude the possibility of p-groups G, such that Σ G is non-trivial, but strictly weaker than the existence of p-cyclic terms.…”