  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 (BLP) and the affine integer
programming relaxation (AIP) is no more powerful than the (in general strictly
weaker) AIP relaxation. Balanced $\mathbf{A}$ include symmetric $\mathbf{A}$
or, more generally, $\mathbf{A}$ preserved by a transitive permutation group.

Computational Complexity;Discrete Mathematics;Data Structures and Algorithms