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

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.