  The promise constraint satisfaction problem (PCSP) is a recently introduced
vast generalisation of the constraint satisfaction problem (CSP) that captures
approximability of satisfiable instances. A PCSP instance comes with two forms
of each constraint: a strict one and a weak one. Given the promise that a
solution exists using the strict constraints, the task is to find a solution
using the weak constraints. While there are by now several dichotomy results
for fragments of PCSPs, they all consider (in some way) symmetric PCSPs.
  1-in-3-SAT and Not-All-Equal-3-SAT are classic examples of Boolean symmetric
(non-promise) CSPs. While both problems are NP-hard, Brakensiek and Guruswami
showed [SODA'18] that given a satisfiable instance of 1-in-3-SAT one can find a
solution to the corresponding instance of (weaker) Not-All-Equal-3-SAT. In
other words, the PCSP template (1-in-3,NAE) is tractable.
  We focus on non-symmetric PCSPs. In particular, we study PCSP templates
obtained from the Boolean template (t-in-k,NAE) by either adding tuples to
t-in-k or removing tuples from NAE. For the former, we classify all templates
as either tractable or not solvable by the currently strongest known algorithm
for PCSPs, the combined basic LP and affine IP relaxation of Brakensiek and
Guruswami [SODA'20]. For the latter, we classify all templates as either
tractable or NP-hard.
