Single Machine Weighted Number of Tardy Jobs Minimization With Small Weights

In this paper we prove new results concerning pseudo-polynomial time algorithms for the classical scheduling problem of minimizing the weighted number of jobs on a single machine, the so-called $1 \mid \mid \Sigma w_j U_j$ problem. The previously best known pseudo-polynomial algorithm for this problem, due to Lawler and Moore [Management Science'69], dates back to the late 60s and has running time $O(d_{\max}n)$ or $O(wn)$, where $d_{max}$ and $w$ are the maximum due date and sum of weights of the job set respectively. Using the recently introduced "prediction technique" by Bateni et al. [STOC'19], we present an algorithm for the problem running in $\widetilde{O}(d_{\#}(n +dw_{\max}))$ time, where $d_{\#}$ is the number of different due dates in the instance, $d$ is the total sum of the $d_{\#}$ different due dates, and $w_{\max}$ is the maximum weight of any job. This algorithm outperform the algorithm of Lawler and Moore for certain ranges of the above parameters, and provides the first such improvement for over 50 years. We complement this result by showing that $1 \mid \mid \Sigma w_j U_j$ has no $\widetilde{O}(n +w^{1-\varepsilon}_{\max}n)$ nor $\widetilde{O}(n +w_{\max}n^{1-\varepsilon})$ time algorithms assuming $\forall \exists$-SETH conjecture, a recently introduced variant of the well known Strong Exponential Time Hypothesis (SETH).