On the Complexity of Symmetric vs. Functional PCSPs

The complexity of the promise constraint satisfaction problem (PCSP) (A, B) is largely unknown, even for symmetric A and B, except for the case when A and B are Boolean. First, we establish a dichotomy for PCSP(A, B) where A, B are symmetric, B is functional (i.e., any r - 1 elements of an r-ary tuple uniquely determines the last one), and (A, B) satisfies technical conditions we introduce called dependency and additivity. This result implies a dichotomy for PCSP(A, B) with A, B symmetric and B functional if (i) A is Boolean, or (ii) A is a hypergraph of a small uniformity, or (iii) A has a relation R A of arity at least three such that the hypergraph diameter of (A, R A ) is at most one. Second, we show that for PCSP(A, B), where A and B contain a single relation, A satisfies a technical condition called balancedness, and B is arbitrary, the combined basic linear programming relaxation and the affine integer programming (AIP) relaxation is no more powerful than the (in general strictly weaker) AIP relaxation. Balanced A include symmetric A or, more generally, A preserved by a transitive permutation group.