Minimizing the Weighted Number of Tardy Jobs via (max,+)-Convolutions

In this paper we consider the fundamental scheduling problem of minimizing the weighted number of tardy jobs on a single machine. We present a simple pseudo polynomial-time algorithm for this problem that improves upon the classical Lawler and Moore algorithm from the late 60’s under certain scenarios and parameter settings. Our algorithm uses (max,+)-convolutions as its main tool, exploiting recent improved algorithms for computing such convolutions, and obtains several different running times depending on the specific improvement used. We also provide a related lower bound for the problem under a variant of the well-known Strong Exponential Time Hypothesis (SETH).

History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete.

Funding: This work was supported by the Israel Science Foundation [Grant 1070/20].