All the stuff that grows in the edges of the beds I chop and compost. The roots aren't hurting anything in the edges because the real crops won't grow well there anyway. Plus I only have a 1/4 acre city lot so I need to make the most out of the compost material I have. I figure it probably helps with keeping the soil on the edges of the beds from getting compacted from me working in the paths.
Anything that makes its way into the beds generally comes up root and all. I leave non-perennial roots in the beds, but the perennial weeds come out.
