That's amazing! Do you have any insight as to which structural properties make this problem simple?

The reason I was asking after tree-width is that control and dataflow graphs in structured programming languages usually have small tree-width (which gets added to the maximum number of overlapping patterns during instruction selection, more or less) and this leads to the fast heuristics used by the PBQP solvers in llvm and libfirm. It's been a few years since I looked into this though, so I'm probably a bit behind the state of the art. :)