I just took a lap around the property spraying the  Air Potato Vines with Round Up – yes, Round Up.  These vines are so invasive and choke out everything else, the native plants, the exotics, everything.  Here’s a great article on Combating Invasive Plants – The Great Air Potato Roundup in Gainesville.  The first couple of years, I simply dug them up and threw them into the campfire.  I’d get hundreds of them.  Once I paid my yard guy’s daughter 10c each for them and let her do the work.  It was worth it.  If I catch them early in the season, and spray the 1.5 solution of Round Up on the emerging tiny leaves as they break the ground, I can nip the invasion in the bud.  I used to just pull up the small plant when I saw the tiny leaves, but, more often than not, that leaves the potato seed in the grow to sprout again.  The big problem the last couple of years is that a neighbor has them and likes them, so they are left to overgrow their fences.  The birds drop seeds over here, too, so my property – the big bird hangout – gets overrun with them.The entire acre on the corner is choked with air potato vines, and I also spray those lots.  Anyway, see this stuff in your yard?  Round Up.  Nothing else works.
UPDATE: This was the last time I used Round Up