The linked GW thread describes one successful approach of which a number of variations have been posted. One I like puts the end of the cut vines into floral tubes of concentrated Roundup.

This didn't totally eliminate the campsis radicans from our badly infested beds, but knocked them back enough so that we could keep ahead of the weak survivors and soil-banked seedlings by hand a few times a year after replanting with other things. Each year they become less of a problem... but it is year 3 of what's looking like a 4 or 5 year journey.

Here is a link that might be useful: GW: Advise please on how to KILL a vine

This post was edited by bostedo on Fri, Apr 18, 14 at 13:35

I've had no trouble killing honeysuckle with Roundup. I buy the concentrate (purple cap) and mix a strong solution. Won't work on ivy, though. It's not all that hard to dig up. Inserting a long root into a bottle of roundup just might work. I've killed bindweed that way.