OMFG kill it with fire!!!
This thing turned up where I lived in Fiji in 1999. We'd never seen it before, but it was called Yaqona-ni-Hawaii or Yaqona-ni-Tonga depending on who you asked, and it was reputed to grow crazy fast. We were kava growers and so of course this plant was intriguing. I procured a stem and planted it in our garden by our porch (a novelty, it would be something to talk about when we had guests over).
So this thing grew and grew and we were amazed and delighted. Something interesting happened as well; it started sprouting up new plants wherever its roots went... kind of like how strawberries throw out runners and grow new plants. Piper Methysticum, unfortunately, doesn't do this. Anyway, it wasn't a problem, the lawnmower took care of any new shoots.
2 years later, it was 8ft tall. Still, the novelty had worn off and I was into a new crop, Noni (aka Kura). So I dug out this Yaqona-ni-Hawaii and planted a noni tree in its place. The branches and roots I tossed down the hill (we lived on the edge of a steep hill).
I leave for a few years and when I go back home, the *entire hillside* is covered... at least three acres of this fucking Piper Auritum!!
Apparently the only way to kill it is to dig it all out of the ground, chop it up into 1" pieces, pour gasoline on it, and then burn it while chanting "burn motherfucker burn" wearing nothing but a loincloth. Glyphosate will also work, but my mom refused to allow any weedicides or pesticides to be used anywhere near our home.
It's still there over a decade later, a source of shame and a constant reminder of the damage invasive species can do.