advice on black swallow wort?

This scourge has been trying to kill everything in our garden, and I just found out this spring that I was using totally wrong methods to get rid of it. For years, I'd pulled it wherever I saw it, and sometimes, where it wasn't too close to rose bushes, etc., I tried digging down to get at the very bottom of roots, but even after digging down at least a foot, and getting to lots of rocks, etc., there was still more root, so at that point I'd give up. This spring at our local flower show, someone explained that when we pull them, we stimulate root growth and they multiply. They suggested cutting instead, so that's what I've begun to do. Also, I got these guys to remove our ugly chain link fence (for free -- they wanted the metal for scrap), so now it's much easier to get at weeds from more angles, so I'm hoping this may be the year we begin to turn it around. Does anyone have other suggestions, besides weekly cutting of this stuff? Our yard seems so small, but we're on a corner, which gives more edges for the black swallow wart to grow along.

I had to look this weed up. My sincere condolences.
Here's the best link on it that I found.
Hope it is of some help.
http://www.dcnr.state.pa.us/forestry/invasivetutorial/black_swallow_wort_M_C.htm

The MUST be a use for this thing.  It stores energy in its roots like nothing else I've seen except sunchokes.  

Probably the easiest solution is shade it out with sunchokes, but not everyone's neighbors appreciate a good sunchoke, right?

It's really hard to find anything but the "control this invasive" type article on the internet, and I found one reference to a moth that eats it and one saying its roots are poisonous to mammals (no references cited for this).  It has some medicinal application in ayurveda, but not especially memorable, and if it's going to grow this abundantly it really wants to be eaten, not just made for a little medicine.  And it seems universally agreed that it is bad for monarch butterflies, not a good substitute for milkweed...but I haven't yet seen a specific reference for this assertion either.

Fish fodder? building biomass under the soil? water retention in the landscape? windbreak? poisoning unfriendly neighbors (just kidding, folks, don't poison your neighbors)?

has anyone tried feeding it to chickens? the root balls, especially?  my friend's quail wouldn't touch them, but chickens seem much more open-minded.