Minimizing the Number of Tardy Jobs and Maximal Tardiness on a Single Machine is NP-hard

TL;DR

This paper proves that minimizing the number of tardy jobs and maximal tardiness on a single machine is strongly NP-hard, resolving a long-standing open problem in multicriteria scheduling.

Contribution

It establishes the NP-hardness of a long-unsolved bicriteria scheduling problem, providing new complexity results for various approaches.

Findings

The problem is strongly NP-hard when minimizing maximal tardiness first.

It is at least weakly NP-hard when minimizing the number of tardy jobs first.

Hardness results are provided for constraint and a priori approaches.

Abstract

This paper resolves a long-standing open question in bicriteria scheduling regarding the complexity of a single machine scheduling problem which combines the number of tardy jobs and the maximal tardiness criteria. We use the lexicographic approach with the maximal tardiness being the primary criterion. Accordingly, the objective is to find, among all solutions minimizing the maximal tardiness, the one which has the minimum number of tardy jobs. The complexity of this problem has been open for over thirty years, and has been known since then to be one of the most challenging open questions in multicriteria scheduling. We resolve this question by proving that the problem is strongly NP-hard. We also prove that the problem is at least weakly NP-hard when we switch roles between the two criteria (i.e., when the number of tardy jobs is the primary criterion). Finally, we provide hardness results for two other approaches (constraint and a priori approaches) to deal with these two criteria.