  We improve the running time of the general algorithmic technique known as
Baker's approach (1994) on H-minor-free graphs from O(n^{f(|H|)}) to O(f(|H|)
n^{O(1)}). The numerous applications include e.g. a 2-approximation for
coloring and PTASes for various problems such as dominating set and max-cut,
where we obtain similar improvements.
  On classes of odd-minor-free graphs, which have gained significant attention
in recent time, we obtain a similar acceleration for a variant of the
structural decomposition theorem proved by Demaine et al. (2010). We use these
algorithms to derive faster 2-approximations; furthermore, we present the first
PTASes and subexponential FPT-algorithms for independent set and vertex cover
on these graph classes using a novel dynamic programming technique.
  We also introduce a technique to derive (nearly) subexponential parameterized
algorithms on H-minor-free graphs. Our technique applies, in particular, to
problems such as Steiner tree, (directed) subgraph with a property, (directed)
longest path, and (connected/independent) dominating set, on some or all proper
minor-closed graph classes. We obtain as a corollary that all problems with a
minor-monotone subexponential kernel and amenable to our technique can be
solved in subexponential FPT-time on H-minor free graphs. This results in a
general methodology for subexponential parameterized algorithms outside the
framework of bidimensionality.
