  In a recent line of work, Butti and Dalmau have shown that a fixed-template
Constraint Satisfaction Problem is solvable by a certain natural linear
programming relaxation (equivalent to the basic linear programming relaxation)
if and only if it is solvable on a certain distributed network, and this
happens if and only if its set of Yes instances is closed under
Weisfeiler-Leman equivalence. We generalize this result to the much broader
framework of fixed-template Promise Valued Constraint Satisfaction Problems.
Moreover, we show that two commonly used linear programming relaxations are no
longer equivalent in this broader framework.
