Did you also know that you can eat kudzu leaves? I don't know why we haven't solved the worlds hunger problem yet but it's true. Kudzu leaves are usually fried when preparing to eat them.

You can also make jam out of it and a mean beer as well.

If you choose to let it grow, just keep it from seeding, and watch out for the taproots, kudzu spreads quickly and efficiently through the use of taproots - it will lay a taproot down and send several runners out in every direction from the taproot, once this process does a rinse/repeat twice, it may very well have taken over your whole yard. Kudzu is programmed to spread far and wide in this manner, and the leaves branch out to create the most possible surface area, allowing it to absorb as much sun as possible shading out anything under it like a tarp - tree, ivy or otherwise - and killing it. The taproots can get to be as big around as a tree in girth.

Personally, if it's possible, I would take Roundup to the whole mess of it. Roundup is incredibly effective in fighting kudzu and Ivy both. I've managed to clear my entire (overgrown) backyard and ditch area of it and it was a total freaking mess when I moved in a couple of years ago. I'm still clearing the last remnants of japanese wisteria (the main root was bigger around than my calves), and Roundup and Brush-B-Gone both seem to work great against all of it.

Just my 2¢.

I read some time ago (w/in last 2 years) that they discovered a fungus that is a natural predator to kudzu, and it spreads fervently (myrothecium verrucaria). We'll never see it used, though, because apparently it attacks cotton as well. :) A quick Google search reveals that a treatment is already patented, and I can't wait till I can get my hands on some.

Here is a link that might be useful: More info on myrothecium verrucaria usage