I have a very short question, but cannot find an answer to it.
Consider the "usual" vertex cover problem, but: (1) It is allowed to select vertices more than once, (2) every edge has a weight $w_e$ attached to it and our vertex cover must "cover" $e$ $w_e$ times, i.e., there must be $w_e$ selected vertices adjacent to it.
In Wikipedia, this is called a "w-vertex cover" (https://en.wikipedia.org/wiki/K%C5%91nig%27s_theorem_(graph_theory) , scroll down to "Weighted variants"), however, this does not seem to be a general term used for this problem.
Thanks in advance!