You can install anything you like, but you will always have root problems - the roots are looking for air, and if concrete hasn't stopped them you can be sure that nothing else will. English Ivy is similarly a thug that has no place in a garden. I would suggest cutting down the tree and replacing it with a more neighbourly variety, and a relentless campaign to remove the ivy. Once those things are done, the patio design is a blank slate. Some barriers to design are meant to be worked around, while others cannot be. KarinL

After cardboard is laid, much of the roots will die. When one of the roots does not die and manages to punch through, your choices are restricted. You can not dig it up without ruining the seal, and if you pull it, there will be more coming from that spot. It is the downside of the very organic method of cardboard+mulch. Yes, touching up with a paint brush is not organic. I also still have more than half of the smallest bottle of Round up you can buy. I bought it in 1998, and I never sprayed it, only applied with a paint brush. With ivy and poison ivy it is a must-have backup. In the case of poison ivy (my #1 enemy) you can not even dig the thing out without risking an ER visit, and the local largest PI vines are (or rather, were) thicker than my forearm and 100 ft long, and a root system to match. After you cut one such thing, no matter how much cardboard and mulch you put down, there will be dozens of shoots punching through within a 20 ft radius for a couple of years.

Significant translocation of glyphosate through soil is rare and most often occurs where excess concentration or over application is used. However, this is a different situation!!! The OP has already pulled up and exposed a bunch or roots. IMO, treating with glyphosate or other herbicide at this point is probably out of the question. Now, I'd probably just use repeated mechanical removal until the ivy gives up, or at least until it is significantly reduced and freshly exposed tree roots are not involved.

Have you found anyone yet? Established Kudzu requires removal of the large Tubers it produces to keep it from returning. This sometimes requires a bit of digging and with many underground utilities a dig permit and utilities location may be required, if it's in the front of your home. This permit is provided free by the utilities companies engineering group upon telephone request. Without it, any damage to utilities(even your own cable tv line if you're with Comcast) could cost you quite a bit of money. I am always in the Buckhead area and would be happy to look at it for you. I have lots of guys that work for me that look for weekend work like this.