Single-Machine Scheduling to Minimize the Number of Tardy Jobs with Release Dates (2408.12967v1)

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.