Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates

TL;DR

The paper investigates the parameterized complexity of the single-machine scheduling problem $1\mid r_j\mid\sum w_j U_j$, which seeks to maximize the weighted count of jobs finishing on time given release dates, due dates, and weights. The authors develop an almost complete complexity map with fixed-parameter tractable algorithms when combining $p_\#$ with any two of $w_\#$, $d_\#$, and $r_\#$, and pseudo-polynomial XP algorithms for fixed $r_\#$ or $d_\#$, while also establishing strong W[1]-hardness for $d_\#+r_\#$ even when $w_\#$ is constant. They achieve these results via reductions to MILP and dynamic programming, constructing both MILP-based FPT algorithms and DP-based XP algorithms, thereby clarifying the tractability frontier across the four parameters. The findings extend the Hermelin et al. results for the release-date-free case and provide a comprehensive framework for understanding how processing-time, weight, due-date, and release-date diversity affect computational complexity in this fundamental scheduling problem. The work has implications for designing efficient algorithms in systems where job availability and deadlines are critical, and it highlights precise parameter regimes where efficient algorithms are possible or unlikely.

Abstract

We study the fundamental scheduling problem $1\mid r_j\mid\sum w_j U_j$: schedule a set of $n$ jobs with weights, processing times, release dates, and due dates on a single machine, such that each job starts after its release date and we maximize the weighted number of jobs that complete execution before their due date. Problem $1\mid r_j\mid\sum w_j U_j$ generalizes both Knapsack and Partition, and the simplified setting without release dates was studied by Hermelin et al. [Annals of Operations Research, 2021] from a parameterized complexity viewpoint. Our main contribution is a thorough complexity analysis of $1\mid r_j\mid\sum w_j U_j$ in terms of four key problem parameters: the number $p_\#$ of processing times, the number $w_\#$ of weights, the number $d_\#$ of due dates, and the number $r_\#$ of release dates of the jobs. $1\mid r_j\mid\sum w_j U_j$ is known to be weakly para-NP-hard even if $w_\#+d_\#+r_\#$ is constant, and Heeger and Hermelin [ESA, 2024] recently showed (weak) W[1]-hardness parameterized by $p_\#$ or $w_\#$ even if $r_\#$ is constant. Algorithmically, we show that $1\mid r_j\mid\sum w_j U_j$ is fixed-parameter tractable parameterized by $p_\#$ combined with any two of the remaining three parameters $w_\#$, $d_\#$, and $r_\#$. We further provide pseudo-polynomial XP-time algorithms for parameter $r_\#$ and $d_\#$. To complement these algorithms, we show that $1\mid r_j\mid\sum w_j U_j$ is (strongly) W[1]-hard when parameterized by $d_\#+r_\#$ even if $w_\#$ is constant. Our results provide a nearly complete picture of the complexity of $1\mid r_j\mid\sum w_j U_j$ for $p_\#$, $w_\#$, $d_\#$, and $r_\#$ as parameters, and extend those of Hermelin et al. [Annals of Operations Research, 2021] for the problem $1\mid\mid\sum w_j U_j$ without release dates.

Theorems & Definitions (28)

• Theorem 1: lenstra1983integer
• Proposition 1: HeegerH24Karp72LenstraRinnooy-Kan77
• Theorem 2
• proof
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Proof of \ref{['thm:fpt1']}