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

2024-04-03

arXiv.org

AI Summary

This paper resolves a longstanding open problem in single-machine scheduling—determining the computational complexity of the bi-criteria scheduling problem under lexicographic optimization, where the primary objective is to minimize maximum lateness (L_{max}) and the secondary objective is to minimize the number of tardy jobs (n_T). We establish, for the first time, that this lexicographic problem is strongly NP-hard. Moreover, we prove that swapping the priority of the two objectives renders the problem only weakly NP-hard. Extending our analysis, we systematically demonstrate NP-hardness under both the weighted-sum (constrained) and preemptive priority (prioritization) approaches. Our proofs rely on careful polynomial-time reductions and multi-criteria scheduling modeling. This work fully characterizes the theoretical hardness boundary of the problem across mainstream multi-objective optimization paradigms, thereby establishing an unattainable complexity benchmark for approximation algorithms and heuristic design.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Resolves complexity of single machine scheduling problem
• Proves problem is strongly NP-hard
• Provides hardness results for multiple approaches

Innovation

• Lexicographic approach for scheduling
• Strongly NP-hard problem resolution
• Hardness results for multiple approaches