Small Promise CSPs that reduce to large CSPs

Abstract

For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A, B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases.If there exists a structure C with homomorphisms A → C → B, then PCSP(A, B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto [2] showed that some PCSPs over finite structures A, B require solving CSPs over infinite C.We show that even when such a reduction to finite C is possible, this structure may become arbitrarily large. For every integer n > 1 and every prime p we give A, B of size n with a single relation of arity n p such that PCSP(A, B) reduces via a chain of homomorphisms A → C → B to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime p ≥ 7 we construct A, B of size p − 1 with a single ternary relation such that PCSP(A, B) reduces via A → C → B to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A, B) reduces to tractable CSP(C) for some finite C, then already A or B has tractable CSP. This extends results and answers a question of [8].