Scheduling with Non-Renewable Resources: Minimizing the Sum of Completion Times

TL;DR

This paper studies scheduling problems with non-renewable resources aiming to minimize total weighted completion times, providing complexity results, approximation algorithms, and a new variant with unknown resource arrivals.

Contribution

It offers polynomial approximation algorithms, proves NP-hardness for specific cases, and introduces a new problem variant with approximation and inapproximability results.

Findings

NP-hardness of unit resource requirement scheduling problem

3/2-approximation using Shortest Processing Time First order

A 4-approximation for the unknown resource arrival times variant

Abstract

The paper considers single-machine scheduling problems with a non-renewable resource. In this setting, we are given a set jobs, each of which is characterized by a processing time, a weight, and the job also has some resource requirement. At fixed points in time, a certain amount of the resource is made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that at any time their resource requirement does not exceed the available amounts of resources. The objective that we consider here is the minimization of the sum of weighted completion times. We give polynomial approximation algorithms and complexity results for single scheduling machine problems. In particular, we show strong NP-hardness of the case of unit resource requirements and weights ($1|rm=1,a_j=1|\sum C_j$), thus answering an open question of Gy\"orgyi and Kis. We also prove that the schedule corresponding to the Shortest Processing Time First ordering provides a $3/2$-approximation for the same problem. We give simple constant factor approximations and a more complicated PTAS for the case of $0$ processing times ($1|rm=1,p_j=0|\sum w_jC_j$). We close the paper by proposing a new variant of the problem in which the resource arrival times are unknown. A $4$-approximation is presented for this variant, together with an $(4-\varepsilon)$-inapproximability result.