Faster Minimization of Tardy Processing Time on a Single Machine

Abstract

This paper is concerned with the $1|| \sum p_j U_j$ 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 an 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 best known running time follows from the famous Lawler and Moore algorithm that solves a more general weighted version 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 problem, each improving on Lawler and Moore’s algorithm in a different scenario.

• Our first algorithm runs in ${\tilde{O}}(P^{7/4})$ time, and outperforms Lawler and Moore’s algorithm in instances where $n={\tilde{\omega }}(P^{3/4})$.

• Our second algorithm runs in ${\tilde{O}}(\min \{P \cdot D_{\#}, P + D\})$ time, where $D_{\#}$ is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore’s algorithm when $n={\tilde{\omega }}(D_{\#})$ or $n={\tilde{\omega }}(D/P)$. Further, it extends the known ${\tilde{O}}(P)$ algorithm for the single due date special case of $1||\sum p_jU_j$ in a natural way.

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, and can be easily extended to the case of a fixed number of machines. For the first algorithm we define a new “skewed” version of $(\max ,\min )$-Convolution which is interesting in its own right.