This plant is every garden owners' worst nightmare. The botanical version of the bubonic plague. So much so that in the UK it is illegal to plant (under the wildlife and countryside act 1981).
If you cut it down, a new plant can re-grow from a segment of stem only a couple of centimeters long and the roots go deep into the ground. Because of this most domestic waste sites WILL NOT accept it, and it has to be taken to an incinerator, treated as hazardous waste.
If you find Japanese knotweed, there's not much you can do
1, spend your next 10 pay-cheques on specialists to remove it
2, try cutting it down yourself, burning the plant completely as you go, and dumping whatever herbicides you can get your hands on all over where it was and hope!
3, build a bonfire on top of the patch..and incinerate it
4, move house and let someone else deal with it