Non-approximability of just-in-time scheduling

Abstract

We consider the following single machine just-in-time scheduling problem with earliness and tardiness costs: Given n jobs with processing times, due dates and job weights, the task is to schedule these jobs without preemption on a single machine such that the total weighted discrepancy from the given due dates is minimum.

NP-hardness of this problem is well established, but no approximation results are known. Using the gap-technique, we show in this paper that the weighted earliness–tardiness scheduling problem and several variants are extremely hard to approximate: If n denotes the number of jobs and b∈ℕ is any given constant, then no polynomial-time algorithm can achieve an approximation which is guaranteed to be at most a factor of O(b ⁿ) worse than the optimal solution unless P = NP.

We also present positive results for two special cases. If the individual processing times and likewise the job weights are similar, i.e., they deviate from each other only by a constant factor, then we obtain constant approximation ratios in pseudopolynomial time. For the second special case, we assume only a constant number of distinct due dates. Then we obtain a pseudopolynomial time exact algorithm using an adaption of a dynamic programming approach introduced by Kolliopoulos and Steiner (Theoretical Computer Science 355:261–273, 2006) for the total weighted tardiness problem.