Primal LP:
min sum_v c_v x_v
s.t.
forall e=vw. x_v + x_w >= 1
forall v. x_v >= 0
Dual LP:
max sum_e y_e
s.t.
forall v. sum_{e=vw} y_e <= c_v
forall e. y_e >= 0
Find a min cut where the edges are arcs from A to B with infinite capacity, the vertices in A are sources, and the vertices in B are sinks, with all vertices having capacity equal to their cost. (Equivalently, make a supersource with arcs to A and a supersink with arcs from B.)
Take the As that are on the "sink" side of the cut and the Bs that are on the "source" side. Every edge vw is covered because if neither v nor w belonged to the cover then vw would be residual.
Hat tip I think to Jenő Egerváry.