  The local search framework for obtaining PTASs for NP-hard geometric
optimization problems was introduced, independently, by Chan and Har-Peled
(2009) and Mustafa and Ray (2010). In this paper, we generalize the framework
by extending its analysis to additional families of graphs, beyond the family
of planar graphs. We then present several applications of the generalized
framework, some of which are very different from those presented to date (using
the original framework). These applications include PTASs for finding a maximum
l-shallow set of a set of fat objects, for finding a maximum triangle matching
in an l-shallow unit disk graph, and for vertex-guarding a
(not-necessarily-simple) polygon under an appropriate shallowness assumption.
  We also present a PTAS (using the original framework) for the important
problem where one has to find a minimum-cardinality subset of a given set of
disks (of varying radii) that covers a given set of points, and apply it to a
class cover problem (studied in Bereg et al., 2012) to obtain an improved
solution.
