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

The 1∣∣ΣwjUj problem asks to determine -- given n jobs each with its own processing time, weight, and due date -- the minimum weighted number of tardy jobs in any single machine non-preemptive schedule for these jobs. This is a classical scheduling problem that generalizes both Knapsack, and Subset Sum. The best known pseudo-polynomial algorithm for 1∣∣ΣwjUj, due to Lawler and Moore [Management Science'69], dates back to the late 60s and has a running time of O(dmaxn), where n is the number of jobs and dmax is their maximal due date. A recent lower bound by Cygan \emph{et al.}~[ICALP'19] for Knapsack shows that 1∣∣ΣwjUj cannot be solved in O˜((n+dmax)2−ε) time, for any ε>0, under a plausible conjecture. This still leaves a gap between the best known lower bound and upper bound for the problem.
In this paper we design a new simple algorithm for 1∣∣ΣwjUj that uses (max,+)-convolutions as its main tool, and outperforms the Lawler and Moore algorithm under several parameter ranges. In particular, depending on the specific method of computing (max,+)-convolutions, its running time can be bounded by
- O˜(n+d#d2max).
- O˜(d#n+d2#dmaxwmax).
- O˜(d#n+d#dmaxpmax).
- O˜(n2+dmaxw2max).
- O˜(n2+d#(dmaxwmax)1.5).
Here, d# denotes the number of \emph{different} due dates in the instance, pmax denotes the maximum processing time of any job, and wmax denotes the maximum weight of any job.