This paper presents an exact algorithm for the identical parallel machine scheduling problem over a formulation where each variable is indexed by a pair of jobs and a completion time.We show that such a formulation can be handled, in spite of its huge number of variables, through a branch cut and price algorithm enhanced by a number of practical techniques, including a dynamic programming procedure to fix variables by Lagrangean bounds and dual stabilization. The resulting method permits the solution of many instances of the P||\sum{wj Tj} problem with up to 100 jobs, and having 2 or 4 machines. This is the first time that medium-sized instances of the P||\sum{wj Tj} have been solved to optimality.