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

We study the fundamental scheduling problem 1 | rj | P wjUj: 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 | rj | P wjUj 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 | rj | P wjUj 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 | rj | P wjUj 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 | rj | P wjUj 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 | rj | P wjUj 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 | rj | P wjUj for p_#, w_#, d_#, and r_# as parameters, and extend those of Hermelin et al. [Annals of Operations Research, 2021] for the problem 1 || P wjUj without release dates.