On Minimizing Tardy Processing Time, Max-Min Skewed Convolution, and Triangular Structured ILPs

The starting point of this paper is the problem of scheduling n jobs with processing times and due dates on a single machine so as to minimize the total processing time of tardy jobs, i.e., (Equation presented). This problem was identified by Bringmann et al. (Algorithmica 2022) as a natural subquadratic-time special case of the classic v problem, which likely requires time quadratic in the total processing time P, because of a fine-grained lower bound. Bringmann et al. obtain their (Equation presented) time scheduling algorithm through a new variant of convolution, dubbed Max-Min Skewed Convolution, which they solve in (Equation presented) time. Our main technical contribution is a faster and simpler convolution algorithm running in (Equation presented) time. It implies an (Equation presented) time algorithm for (Equation presented), but may also be of independent interest. Inspired by recent developments for the Subset Sum and Knapsack problems, we study (Equation presented) parameterized by the maximum job processing time pmax. With proximity techniques borrowed from integer linear programming (ILP), we show structural properties of the problem that, coupled with a new dynamic programming formulation, lead to an (Equation presented) time algorithm. Moreover, in the setting with multiple machines, we use similar techniques to get an (Equation presented) time algorithm for (Equation presented). Finally, we point out that the considered problems exhibit a particular triangular block structure in the constraint matrices of their ILP formulations. In light of recent ILP research, a question that arises is whether one can devise a generic algorithm for such a class of ILPs. We give a negative answer to this question: we show that already a slight generalization of the structure of the scheduling ILP leads to a strongly NP-hard problem.