Cooperative Extension: Garden and Yard

The Penn State Univ Extension has an excellent page on the plant, including advice on how to control it. The State of Maine has a similar page. The first step, regardless of intervention method, is to sever the vine from its roots (if it’s climbing a tree, two cuts one at ground level and another at eye level, are recommended to make sure you’ve separated it). Do not try to pull down the existing vines as that tends to cause unnecessary damage to the tree. Severed vines will wither, dry, and decompose on their own.The root stock is the bigger problem. Generally, chemical intervention is the preferred method at that point, but I understand and respect your efforts to keep away from pesticides. What you’re then looking at is conscientious removal of new shoots as they emerge from the root stock; it will take several seasons, but with diligence you can eventually starve out the underground portions of the plant. Take special care to inspect the area around the roots/new shoots, as the plant can spread underground and you want to make sure it isn’t sneaking a shoot up a nearby tree without you noticing.