This paper is concerned with the $1||\sum p_jU_j$ problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in $O(P \cdot n)$ time, where $P$ is the total processing time of all $n$ jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date.
In this paper we develop two new algorithms for $1||\sum p_jU_j$, each improving on Lawler and Moore's algorithm in a different scenario. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of $(\max,\min)$-convolution which is interesting in its own right.
Rethinking Warm-Starts with Predictions: Learning Predictions Close to Sets of Optimal Solutions for Faster $\text{L}$-/$\text{L}^\natural$-Convex Function Minimization
Faster Exponential Algorithms for Multi-Machine Scheduling Problems
Max Planck Institute for Informatics（马克斯·普朗克计算机科学研究所）；Saarland University（萨尔兰大学）；ETH, Zurich（苏黎世联邦理工学院）；CISPA Helmholtz Center for Information Security（CISPA 赫尔姆霍兹信息安全中心）
