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An overview of some single machine scheduling problems: polynomial algorithms, complexity and approximability

Nodari Vakhania, Frank Werner, Kevin Johedan Ramírez-Fuentes, Víctor Pacheco-Valencia

TL;DR

This paper surveys polynomial-time algorithms, complexity results, and approximation approaches for single-machine scheduling problems, emphasizing results from recent decades and the authors’ contributions. It surveys how release times, due dates, tails, and preemption affect tractability, covering topics from $1|r_j|L_{ m max}$ and $ ext{sum }U_j$ to tardiness and resource-constrained variants, with key reductions and algorithmic techniques such as implicit enumeration, dynamic programming, and graphical methods. The work highlights both strong NP-hardness in general and a breadth of polynomial-time solvable special cases, offering structural insights and strategies that inform more complex multi-machine and resource-aware scheduling. The findings have practical impact by enabling efficient lower bounds, exact methods for benchmark subproblems, and principled guidelines for extending single-machine results to broader scheduling contexts.

Abstract

Since the publication of the first scheduling paper in 1954, a huge number of works dealing with different types of single machine problems appeared. They addressed many heuristics and enumerative procedures, complexity results or structural properties of certain problems. Regarding surveys, often particular subjects like special objective functions are discussed, or more general scheduling problems were surveyed, where a substantial part is devoted to single machine problems. In this paper we present some results on polynomial algorithms, complexity and approximation issues, where the main focus is on results, which have been published during the last decades in papers, where at least one of the first two authors of this paper was involved. We hope that the reviewed results will stimulate further investigation in related research fields.

An overview of some single machine scheduling problems: polynomial algorithms, complexity and approximability

TL;DR

This paper surveys polynomial-time algorithms, complexity results, and approximation approaches for single-machine scheduling problems, emphasizing results from recent decades and the authors’ contributions. It surveys how release times, due dates, tails, and preemption affect tractability, covering topics from and to tardiness and resource-constrained variants, with key reductions and algorithmic techniques such as implicit enumeration, dynamic programming, and graphical methods. The work highlights both strong NP-hardness in general and a breadth of polynomial-time solvable special cases, offering structural insights and strategies that inform more complex multi-machine and resource-aware scheduling. The findings have practical impact by enabling efficient lower bounds, exact methods for benchmark subproblems, and principled guidelines for extending single-machine results to broader scheduling contexts.

Abstract

Since the publication of the first scheduling paper in 1954, a huge number of works dealing with different types of single machine problems appeared. They addressed many heuristics and enumerative procedures, complexity results or structural properties of certain problems. Regarding surveys, often particular subjects like special objective functions are discussed, or more general scheduling problems were surveyed, where a substantial part is devoted to single machine problems. In this paper we present some results on polynomial algorithms, complexity and approximation issues, where the main focus is on results, which have been published during the last decades in papers, where at least one of the first two authors of this paper was involved. We hope that the reviewed results will stimulate further investigation in related research fields.
Paper Structure (14 sections, 1 theorem, 7 equations)

This paper contains 14 sections, 1 theorem, 7 equations.

Table of Contents

  1. Introduction
  2. Some Basics
  3. Polynomially Solvable Cases, Complexity and Approximability Issues
  4. Scheduling a Single Machine with Release Times and Due Dates to Minimize Maximum Job Lateness
  5. Minimizing the Number of Late Jobs
  6. Nonpreemptive Scheduling to Minimize the Number of Late Jobs: $1|r_j;p_j=p|\sum U_j$
  7. Preemptive Scheduling to Maximize the Number of on Time Jobs: $1|r_j;pmtn|\sum U_j$
  8. Preemptive Scheduling to Minimize the Weighted Number of Late Jobs with Equal Length: $1|r_j; p_j = p; pmtn|\sum w_j U_j$
  9. Scheduling to Minimize the Weighted Number of Late jobs with Deadlines and Release/Due Date Intervals
  10. Scheduling with Tardiness-based Objectives
  11. Preemptive Scheduling in Overloaded Systems
  12. Scheduling with Restarts
  13. Scheduling with Financial Resource Constraints
  14. Concluding Remarks

Key Result

Lemma 2.1

Let $S$ be a strongly C-schedule and let $l$ be the emerging job scheduled in $i$th position in in that schedule. Let, further $j$ be the job, immediately succeeding job $l$ in schedule $S$. Then if $d_j < d_l$, there will occur a gap in between the $i$th scheduled job and the kernel $K(S)$ in the s

Theorems & Definitions (1)

  • Lemma 2.1