We consider a variety of preemptive scheduling problems with controllable processing times on a single machine and on identical/uniform parallel machines, where the objective is to minimize the total compression cost. In this paper, we propose fast divide-and-conquer algorithms for these scheduling problems. Our approach is based on the observation that each scheduling problem we discuss can be formulated as a polymatroid optimization problem. We develop a novel divide-and-conquer technique for the polymatroid optimization problem and then apply it to each scheduling problem. We show that each scheduling problem can be solved in O(Tfeas(n)× log n) time by using our divide-and-conquer technique, where n is the number of jobs and O(T feas(n) denotes the time complexity of the corresponding feasible scheduling problem with n jobs. This approach yields faster algorithms for most of the scheduling problems discussed in this paper.