Die, Houttuynia, die!
I've just been reading the thread "Help! Chameleon plant (Houttuynia cordata 'Chameleon') taking over" and maybe, just maybe, I might have a solution (This year will tell). But first you have to realize how unbelievably invasive this monster is (and yes, I am the one who planted it, years ago).
Previously, I tried everything, including Roundup (It just laughed, the leaves turned crispy and brown, and in no time, new leaves grew back). I think I reached the point of tears when I tried to dig out just one cubic foot of soil infested with the tangled-spaghetti-like roots, and easily filled up half a huge Rubbermaid garbage can just from that small space... and could have gotten more if I dug deeper (not to mention all around the dig-site).
OK, so I thought, what else do I have that's invasive, but not so hard to remove? You'll never believe it: Italian musk strawberries! I once bought a small number of these from Raintree Nurseries, thinking they'd stay in a 3 x 3 ft. raised-bed garden. In no time they climbed over and out and were everywhere, just about growing on a nearby concrete patio extension. But they were easy to pull out, so I could keep yanking them.
So, the year before last, I deliberately planted some of these strawberries among the Houttuynia in the area where this quicksilver-fast-spreader has taken over. (It's a triangular section 17 x 17 x 19 ft bordered on all three sides by a low stone wall and it has two mini-drawf apples trees there). I only put the strawberries in the back, after first trying to clear out an area of surface Houttuynia (there is no going deep enough to get rid of all the roots). Well, last year was the test, and by the end of summer, although the Houttuynia was still there, it was actually rather well-behaved in the back, not growing to its "usual" height (in my sprinkler-system watered area) of two feet tall.
So here's the hopeful news at the beginning of spring. I just raked out the entire triangular area (the Houttuynia isn't up yet, though the tips of the new season's growth are showing) - and now that I've raked up the leaves, I can see that the strawberries have grown to cover the complete triangular area! They may never kill the Houttuynia, but I don't care as long as there's a chance that they will keep it from ever again growing two feet tall. And maybe eventually the strawberries will completely win over the Houttuynia!?
I'd be interested if anyone knows how/why this seems to be working. Planting a few Italian musk strawberries was certainly MUCH easier than trying to bring in a bulldozer to a stone wall-surrounded area where a dozer wouldn't work anyway. I don't know if this is the only kind of competitor-plant that would work; they were simply the handiest invasive-but-not-deep-rooted plant I happened to have at hand.