Well - you have your knotweed - - around here its Morning Glory/Bind weed. Both are a tough, invasive customer. Both spread from rhizomes.
I have a pasture that we first though was - "Oh, so pretty". Then the rancher, neighbor, came over and told us what that was and that we better start doing something or it could take over the entire eighty acres.
Literature suggests that digging up the root balls is a waste of time and actually spreads the plant. Every tiny little bit of root generates a new plant.
The suggestion is RoundUP early in the spring when the plant is just starting to emerge.
I have reduced the size of the "infested area" from a quarter acre down to the size of your kitchen table. This will be the fourth year of my active attack against this plant. I can't believe how easily this plant spreads.
I seriously doubt that I will ever totally eliminate the Bind weed - but it will be down to the size that is easy to maintain and control.