Computational Complexity of Minimizing Tardy Jobs and Maximum Tardiness on a Single Machine

Core Concepts

The problem of minimizing the number of tardy jobs while also minimizing the maximum tardiness on a single machine is strongly NP-hard.

Abstract

The paper resolves the long-standing open question on the computational complexity of single machine bicriteria scheduling problems involving the number of tardy jobs (PUj) and the maximum tardiness (Tmax) criteria.
The key highlights and insights are:

The authors prove that the constrained problem 1|Tmax ≤ℓ, PUj ≤k| is strongly NP-complete, by a reduction from the 3-Partition problem.

They show that the lexicographic problem 1||Lex(Tmax, PUj) and the a priori problem 1||αTmax + PUj are both strongly NP-hard, as direct consequences of the hardness result for the constrained problem.

For the lexicographic problem 1||Lex(PUj, Tmax), the authors devise a reduction from the weakly NP-complete Partition problem, establishing its weak NP-hardness.

The paper resolves the complexity status of all single machine bicriteria scheduling problems involving the Tmax and PUj criteria, settling the long-standing open question in the literature.

Stats

None.

Quotes

None.

Key Insights Distilled From

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

by Klaus Heeger... at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02784.pdf

Deeper Inquiries

What are some practical applications of the single machine bicriteria scheduling problems involving Tmax and PUj

The single machine bicriteria scheduling problems involving Tmax and PUj have practical applications in various industries. One application is in manufacturing, where a production facility needs to schedule jobs on a single machine to minimize both the maximum tardiness of jobs and the total number of tardy jobs. This can help optimize production schedules to ensure timely delivery of products and reduce the overall number of delays. Another application is in service industries, such as healthcare or transportation, where scheduling appointments, surgeries, or routes on a single machine needs to balance between minimizing the maximum tardiness and the total number of tardy jobs to improve efficiency and customer satisfaction.

Can the hardness results be extended to more general machine environments beyond the single machine case

The hardness results obtained for the single machine bicriteria scheduling problems involving Tmax and PUj can potentially be extended to more general machine environments beyond the single machine case. By adapting the reduction techniques and complexity analysis used in the proof of NP-hardness for the single machine case, similar results can be established for multiple machine environments. The key lies in formulating the problem constraints, objectives, and job characteristics in a way that captures the complexity of scheduling on multiple machines while considering both the maximal tardiness and the number of tardy jobs.

Are there any approximation algorithms or heuristics that can provide good solutions for these NP-hard scheduling problems in practice

While the single machine bicriteria scheduling problems involving Tmax and PUj are NP-hard, there are approximation algorithms and heuristics that can provide good solutions in practice. One approach is to develop approximation algorithms that provide solutions with provable performance guarantees, such as approximation ratios or bounds on the quality of the solution compared to the optimal solution. Additionally, heuristic methods, such as genetic algorithms, simulated annealing, or tabu search, can be employed to find near-optimal solutions efficiently. These methods may sacrifice optimality for computational efficiency but can still yield practical solutions for real-world scheduling problems.