## Abstract

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 ||∑

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 || PwjUj has no Oe(n + w 1−ε maxn) nor Oe(n + wmaxn 1−ε ) time algorithms assuming ∀∃-SETH conjecture, a recently introduced variant of the well known Strong Exponential Time Hypothesis (SETH).

_{wj}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 dmax and ω 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 Oe(d#(n+dwmax)) 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 wmax is the maximum weight of any job. This algorithm outperform thealgorithm 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 || PwjUj has no Oe(n + w 1−ε maxn) nor Oe(n + wmaxn 1−ε ) time algorithms assuming ∀∃-SETH conjecture, a recently introduced variant of the well known Strong Exponential Time Hypothesis (SETH).

Original language | English |
---|---|

State | Published - 2022 |