  We study polynomial-time approximation schemes (PTASes) for constraint
satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex
Cover on sparse graph classes.
  Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and
beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs
(where constraints are arbitrary non-negative functions), Romero, Wrochna, and
\v{Z}ivn\'y [SODA'21] showed that the Sherali-Adams LP relaxation gives a
simple PTAS for all fractionally-treewidth-fragile classes, which is the most
general "sparsity" condition for which a PTAS is known. We extend these results
to general-valued CSPs, which include "crisp" (or "strict") constraints that
have to be satisfied by every feasible assignment. The only condition on the
crisp constraints is that their domain contains an element which is at least as
feasible as all the others (but possibly less valuable).
  For minimisation general-valued CSPs with crisp constraints, we present a
PTAS for all Baker graph classes -- a definition by Dvo\v{r}\'ak [SODA'20]
which encompasses all classes where Baker's technique is known to work, except
possibly for fractionally-treewidth-fragile classes. While this is standard for
problems satisfying a certain monotonicity condition on crisp constraints, we
show this can be relaxed to diagonalisability -- a property of relational
structures connected to logics, statistical physics, and random CSPs.
