Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Klaus Heeger, Danny Hermelin, Michael L. Pinedo, Dvir Shabtay

TL;DR

The paper addresses the complexity of bicriteria single-machine scheduling with the primary objective $T_{\max}$ and the secondary objective $\sum U_j$ under lexicographic, constraint, and a priori formulations. By constructing a large-parameter, period-based reduction from 3-Partition, it proves that $1|T_{\max} \le \ell, \sum U_j \le k|$ is strongly NP-complete and that the lexicographic problem $1||Lex(T_{\max},\sum U_j)$ is also strongly NP-complete, while $1||Lex(\sum U_j, T_{\max})$ is only weakly NP-complete via a Partition reduction. It further shows that the a priori variant $1||\alpha T_{\max}+\sum U_j$ is strongly NP-complete. Collectively, these results settle long-standing open questions in multicriteria single-machine scheduling and delineate sharper hardness boundaries for related approaches, such as constraint formulations and a priori combinations, with implications for potential PTAS or FPT explorations in future work.

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.

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

TL;DR

The paper addresses the complexity of bicriteria single-machine scheduling with the primary objective and the secondary objective under lexicographic, constraint, and a priori formulations. By constructing a large-parameter, period-based reduction from 3-Partition, it proves that is strongly NP-complete and that the lexicographic problem is also strongly NP-complete, while is only weakly NP-complete via a Partition reduction. It further shows that the a priori variant is strongly NP-complete. Collectively, these results settle long-standing open questions in multicriteria single-machine scheduling and delineate sharper hardness boundaries for related approaches, such as constraint formulations and a priori combinations, with implications for potential PTAS or FPT explorations in future work.

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.
Paper Structure (16 sections, 23 theorems, 61 equations, 2 tables)

This paper contains 16 sections, 23 theorems, 61 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Preliminaries
  5. Strong NP-completeness
  6. Construction
  7. Job construction.
  8. Parameters $\boldmath{k}$ and $\boldmath{\ell}$.
  9. Proof roadmap.
  10. Job sets $\boldmath{\mathcal{J}_{\le j}}$ and $\boldmath{\mathcal{J}^*_{\le j}}$
  11. Candidate Sets
  12. Correctness
  13. Proof of Lemma \ref{['lem:only-candidate']}
  14. Lexicographically first minimizing $T_{\max}$ and then $\sum U_j$
  15. Lexicographically first minimizing $\sum U_j$ and then $T_{\max}$
  16. ...and 1 more sections

Key Result

Theorem 1

$1|T_{\max} \le \ell, \sum U_j \le k|$ is strongly NP-complete.

Theorems & Definitions (52)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Definition 1: Canonical schedule
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • Definition 2
  • ...and 42 more