I have fought wild blackberries here in the city. Our realtor thought they were poison ivy, but I told here no, just wild blackberries. They are growing underneath my privet hedges out front but not all that bad really. From the reading I've done on them, most sources say to use both mechanical removal (digging up small younger shoots) and chemical application.
I can tell you from experience thus far, starving them from light does not kill them. The vines/stems will turn white from lack of light, but they will just keep on running out farther and farther, above or below ground, making new plants. So far I have reduced the quantity of vines out front by remaining watchful and persistent pulling/digging up of shoots as soon as they emerge from the dirt. A compounding problem for us is our house is pier-and-beam up off the ground 2', so the runners creep through the metal vents in the brick skirting and have gone under the house to our side yard! My tact so far is to just pull up by hand any new shoots that emerge into the side garden the minute I see one emerge. So far, that is working pretty well to keep them in check, though not totally eradicated. We can't (or perhaps I should say neither my husband nor I are WILLING to get in that crawl space) & deal with them there. So we'll just keep doing what we are doing.
My late brother had them real bad on his place in Tulalip, WA and used some sort of John Deere brush cutter to get the largest clumps of them under control, but again, he never totally eliminated them. They were kept in check enough you could at least walk the back yard in your bare feet.