Improved dynamic programs for batching problems with maximum lateness criterion

We study a class of scheduling problems involving the maximum lateness criterion and an element of batching. For all the problems that we examine, algorithms appear in the literature which consist of a sorting step to determine an optimal job sequence, followed by a dynamic programming step which determines the optimal batches. In each case, the dynamic program is based on a backward recursion of which a straightfoward implementation requires O(n^2) time, where n is the number of jobs. We present improved implementations of these dynamic programs that are based on monotonicity properties of the objective expressed as a function of the length of the first batch. These properties and the use of efficient data structures enable us to exclude partial schedules that cannot lead to an overall optimum early on in the enumeration process. The four problems that we consider are solved in O(n log n) time; in two occasions, the batching step is actually performed in linear time and the overall complexity is determined by the sorting step.