Scheduling jobs with equal processing times on a single machine: minimizing maximum lateness and makespan

Abstract

The following special case of the classical NP-hard scheduling problem $1|r_j|L_{\max }$ is considered. There is a set of jobs $N= \{ 1, 2, \ldots , n \}$ with identical processing times $p_j=p$ for all jobs $j \in N$. All jobs have to be processed on a single machine. The optimization criterion is the minimization of maximum lateness $L_{\max }$. We analyze algorithms for the makespan problem $1|r_j|C_{\max }$, presented by Garey et al. (SIAM J Comput 10(2):256–269, 1981), Simons (A fast algorithm for single processor scheduling. In: 19th Annual symposium on foundations of computer science (Ann Arbor, Mich., 1978, 1978) and Benson’s algorithm (J Glob Optim 13(1):1–24, 1998) and give two polynomial algorithms to solve the problem under consideration and to construct the Pareto set with respect to the criteria $L_{\max }$ and $C_{\max }$. The complexity of the presented algorithms is $O(Q \cdot n \log n )$ and $O(n^3 \log n)$, respectively, where $10^{-Q}$ is the accuracy of the input-output parameters.