On the complexity of symmetric vs. functional PCSPs

The complexity of the promise constraint satisfaction problem \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) is largely unknown, even for symmetric \(\mathbf{A}\) and \(\mathbf{B}\), except for the case when \(\mathbf{A}\) and \(\mathbf{B}\) are Boolean.

First, we establish a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) where \(\mathbf{A},\mathbf{B}\) are symmetric, \(\mathbf{B}\) is functional (i.e. any \(r-1\) elements of an \(r\)-ary tuple uniquely determines the last one), and \((\mathbf{A},\mathbf{B})\) satisfies technical conditions we introduce called dependency and additivity. This result implies a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) with \(\mathbf{A},\mathbf{B}\) symmetric and \(\mathbf{B}\) functional if (i) \(\mathbf{A}\) is Boolean, or (ii) \(\mathbf{A}\) is a hypergraph of a small uniformity, or (iii) \(\mathbf{A}\) has a relation \(R^{\mathbf{A}}\) of arity at least 3 such that the hypergraph diameter of \((A,R^{\mathbf{A}})\) is at most 1.

Second, we show that for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\), where \(\mathbf{A}\) and \(\mathbf{B}\) contain a single relation, \(\mathbf{A}\) satisfies a technical condition called balancedness, and \(\mathbf{B}\) is arbitrary, the combined basic linear programming relaxation (\(\operatorname{BLP}\)) and the affine integer programming relaxation (\(\operatorname{AIP}\)) is no more powerful than the (in general strictly weaker) \(\operatorname{AIP}\) relaxation. Balanced \(\mathbf{A}\) include symmetric \(\mathbf{A}\) or, more generally, \(\mathbf{A}\) preserved by a transitive permutation group.