  We show for a broad class of counting problems, correlation decay (strong
spatial mixing) implies FPTAS on planar graphs. The framework for the counting
problems considered by us is the Holant problems with arbitrary constant-size
domain and symmetric constraint functions. We define a notion of regularity on
the constraint functions, which covers a wide range of natural and important
counting problems, including all multi-state spin systems, counting graph
homomorphisms, counting weighted matchings or perfect matchings, the subgraphs
world problem transformed from the ferromagnetic Ising model, and all counting
CSPs and Holant problems with symmetric constraint functions of constant arity.
  The core of our algorithm is a fixed-parameter tractable algorithm which
computes the exact values of the Holant problems with regular constraint
functions on graphs of bounded treewidth. By utilizing the locally tree-like
property of apex-minor-free families of graphs, the parameterized exact
algorithm implies an FPTAS for the Holant problem on these graph families
whenever the Gibbs measure defined by the problem exhibits strong spatial
mixing. We further extend the recursive coupling technique to Holant problems
and establish strong spatial mixing for the ferromagnetic Potts model and the
subgraphs world problem. As consequences, we have new deterministic
approximation algorithms on planar graphs and all apex-minor-free graphs for
several counting problems.
