A dynamic programming algorithm for preemptive scheduling of a single machine to minimize the number of late jobs

The scheduling problem 1|pmtn, r _j |w _j U _j calls forn jobs with arbitrary release dates and due dates to be preemptively scheduled for processing by a single machine, with the objective of minimizing the sum of the weights of the late jobs. A dynamic programming algorithm for this problem is described. Time and space bounds for the algorithm are, respectively,O(nk ² W ²) andO(k ² W), wherek is the number of distinct release dates andW is the sum of the integer job weights. Thus, for the problem 1|pmtn, r _j |U _j , in which the objective is simply to minimize the number of late jobs, the pseudopolynomial time bound becomes polynomial, i.e.O(n ³ k ²).