I've found a bunch of websites from the UK and the US detailing various control strategies - organic, chemical, and large scale. I should be able to take enough information from those to deal with this.
Here's the really awful part - I think this stuff was planted intentionally! There are a few varieties of polygonum that are non-invasive, but this one is downright thuggish, and it's still sold as groundcover. People are stupid.
Now I'm afraid to dispose of *anything* I have to dig up...