I have a lot of experience getting rid of both Kudzu and Bamboo. It's easier to get rid of Kudzu. You spray it with Round-up. The areas that don't die or come back, you spray again. It may require a few treatments and severing of the larger parts of the vines.
Bamboo on the other hand requires a different strategy. Round-up does very little on large bamboo. You first have to cut it all down. After that, what comes up after that within the first year are small shoots with a lot of low growing leaves. Round-Up does work on these small shoots. It also requires multiple applications of Round-up over the summer but it will be all but gone by the next year. You will have a some new small shoots pop up the next year but another dose or two of Round-up will knock them out. The big difference with bamboo is dealing with the large stumps left behind after they are cut down. The larger ones take several years to rot enough to kick over. It's ideal to cut them as low as possible so you can start mowing the area. Cutting all of it down and regular mowing the area causes the rhizomes to lose their energy.