  The primal-dual scheme has been used to provide approximation algorithms for
many problems. Goemans and Williamson gave a (2-1/(n-1))-approximation for the
Prize-Collecting Steiner Tree Problem that runs in O(n^3 log n) time. it
applies the primal-dual scheme once for each of the n vertices of the graph.
Johnson, Minkoff and Phillips proposed a faster implementation of Goemans and
Williamson's algorithm. We give a proof that the approximation ratio of this
implementation is exactly 2.
