Throughput Scheduling with Equal Additive Laxity

Abstract

We study a special case of a classical throughput maximization problem. There is given a set of jobs, each job j having a processing time $p_j$, a release time $r_j$, a deadline $d_j$, and possibly a weight. The jobs have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule for a subset of jobs of maximum cardinality (or maximum total weight) that start and finish within their feasible time window $[r_j,d_j)$. In our special case, the additive laxity of every job is equal, i.e., $d_j-p_j-r_j= \delta$ with a common $\delta$ for all jobs. Throughput scheduling has been studied extensively over decades. Understanding important special cases is of major interest. From a practical point of view, our special case was raised as important in the context of last-mile meal deliveries. As a main result we show that single-machine throughput scheduling with equal additive laxity can be solved optimally in polynomial time. This contrasts the strong NP-hardness of the problem variant with arbitrary (and even equal multiplicative) laxity. Further, we give a fully polynomial-time approximation scheme for the weakly NP-hard weighted problem. Our single-machine algorithm can be used repeatedly to schedule jobs on multiple machines, such as in the greedy framework by Bar-Noy et al. [STOC ’99], with an approximation ratio of $\frac{(m)^m}{(m)^m - (m-1)^m}\!<\!\frac{e}{e-1}$. Finally, we present a pseudo-polynomial time algorithm for our weighted problem on a constant number of machines.