  While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin,
Guruswami, and H\r{a}stad roved a result known as "$(2+\varepsilon)$-SAT is
NP-hard" [FOCS'14/SICOMP'17]. They showed that the problem of distinguishing
k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least
g literals in every clause) from those that are not even 1-satisfiable is
NP-hard if $\frac{g}{k} < \frac{1}{2}$ and is in P otherwise. We study a
generalisation of SAT on arbitrary finite domains, with clauses that are
disjunctions of unary constraints, and establish analogous behaviour. Thus we
give a dichotomy for a natural fragment of promise constraint satisfaction
problems (PCSPs) on arbitrary finite domains.
  The hardness side is proved using the algebraic approach, via a new general
NP-hardness criterion on polymorphisms of the problem, based on a gap version
of the Layered Label Cover problem. We show that previously used criteria are
insufficient -- the problem hence gives an interesting benchmark of algebraic
techniques for proving hardness of approximation problems such as PCSPs.
