Might this Potato Vine be the "Solanum laxum"?
This is hardy to about Plant Hardiness Zone 9 so theoretically anything colder should kill it. Where in the United States are you? Mid south is not enough information.
Northern Georgia, southern Aalabama, eastern South Carolina would be better.

You may be able to use the method described here by Ken.

I am in North Carolina. This vine is not the bright lime green type you see in stores now, it is dark green and has the little potatoes on the vine. It comes back every year no matter what! The small shoots are coming up everywhere. I have tried to dig up the larger vines but there are runners, so even if I get the large vine, a couple of months later I see the 'baby vines' coming up also.

I think you are talking about Dioscorea bulbifera (air potato.)

Have you taken a sample to your local North Carolina State University Cooperative Extension Service office for identification and suggestions on how to control this?

Here is a link that might be useful: NCSU CES

{{gwi:27445}}
If you have had it for 20 years, it is going to be hard to get rid of. Round up will kill it, but you would have to keep after it. I have given up ever getting rid of cinnamon vine which is what I think you have. There are no kids around here I live in the country. I can put very dilute roundup in a bottle and put the tips of vine in the bottle. It would take forever. Concentrated round up in the bottle would kill just the end of the vine so use a dilute mixture.

Usually the best time of year to use weed killer is when the plant is actively growing. You could lay down cardboard around your good plants to suppress the top growth and maybe eventually rot the bad plant. Find out exactly what your enemy is by calling your county extension office. Don't let it ruin your fun in the garden!

The cardboard is a good idea. This stuff, when I pulled it up, left the little potatoes all through the garden, so now I have more coming up. I can't stand the idea that I might have to dig up all the other plants to get rid of it - and then I might miss some of the roots and it would come back up after all the work. It is strangling a small crepe myrtle and is in among the phlox - I keep pulling it up but smaller ones come up nearby.

I keep seeing post from people that say a glyphosate product will kill this weed but it keeps growing back. That means the plant poison used did not really work. The researchers at Iowa State University have found that more and more of what we call "weeds" are developing immunity to the glyphosate products. That tells me that someting that is very expensive does not work as well as the ads suggest.