Overarching reasons why problems are in P or BPP

Recently, when talking to a physicist, I claimed that in my experience, when a problem that naively seems like it should take exponential time turns out nontrivially to be in P or BPP, an "overarching reason" why the reduction happens can typically be identified---and almost always, that reason belongs to a list of a dozen or fewer "usual suspects" (for example: dynamic programming, linear algebra...). However, that then got me to thinking: can we actually write down a decent list of such reasons? Here's a first, incomplete attempt at one: (0) Mathematical characterization. Problem has a non-obvious "purely-mathematical" characterization that, once known, makes it immediate that you can just do exhaustive search over a list of poly(n) possibilities. Example: graph planarity, for which an O(n 6 ) algorithm follows from Kuratowski's theorem. (As "planar" points out below, this was a bad example: even once you know a combinatorial characterization of planarity, giving a polynomial-time algorithm for it is still quite nontrivial. So, let me substitute a better example here: how about, say, "given an input n written in binary, compute how many colors are needed to color an arbitrary map embedded on a surface with n holes." It's not obvious a priori that this is computable at all (or even finite!). But there's a known formula giving the answer, and once you know the formula, it's trivial to compute in polynomial time. Meanwhile, "reduces to excluded minors / Robertson-Seymour theory" should probably be added as a separate overarching reason why something can be in P.) Anyway, this is specifically not the sort of situation that most interests me. (1) Dynamic programming. Problem can be broken up in a way that enables recursive solution without exponential blowup -- often because the constraints to be satisfied are arranged in a linear or other simple order. "Purely combinatorial"; no algebraic structure needed. Arguably, graph reachability (and hence 2SAT) are special case