Where you have an established invasive in your walk ways, I think it would be important to spend at least one season eradicating it from the walk ways, and keep removing it from the beds as it shows up. The choice re: round up is only one that you can make. If I were dealing with this situation in my yard, I'd heavily mulch the walk ways with cardboard or news paper covered with a layer of straw or shavings or wood chips. Or you could cover with old carpet (least desirable IMO) to exclude light and kill off the main growth of plants. Then go after the weeds in the garden beds with heavy weeding followed by mulch to exclude light from the soil. Any weed growth that breaks through should be lighter and easier to remove. A tiller may only break up the root system and encourage more vegetative propogation. (I'm not familiar with your specific weed) I don't think there is an easy fast way to get rid of an invasive, but heavy mulching and persistence should accomplish the task over several seasons. Let the mulch do the work for you.