Among other things, I have to do something with the blackberry vines. They grow and grow and grow, and then when they reach the ground, a root forms at the tip of the vine. And that root sprouts more vines. It's so easy for them to infest an area, and difficult for us to figure out where it all begins.
Chainsaws can be useful to clear a lot away, but aren't accessible or advisable for everyone. Try to cut it down with clippers and you'll spend all your time defending the ground you've gained from new growth.
Far better to put on thick gloves and pull. Tear at it until it rips out, like old fashioned wires from the walls of a cheap house. And there it will be - the new root that has buried itself deep down in the ground. Cut it off and look for another. Repeat until the garden is clear.
It takes a lot of work, and it isn't an instant fix. But the more roots pulled out, the less work maintaining it in the future, and the healthier the garden will be.