Split Scheduling with Uniform Setup Times

TL;DR

This paper investigates split scheduling with uniform setup times, providing polynomial algorithms for two machines, approximation schemes for multiple machines, and establishing NP-hardness for weighted completion time minimization.

Contribution

It introduces a polynomial-time algorithm for total completion time minimization on two machines and approximation algorithms for multiple machines, addressing a practical scheduling problem with setup times.

Findings

Polynomial-time algorithm for two-machine total completion time minimization.

Constant-factor approximation for multiple machines.

NP-hardness for weighted total completion time problem.

Abstract

We study a scheduling problem in which jobs may be split into parts, where the parts of a split job may be processed simultaneously on more than one machine. Each part of a job requires a setup time, however, on the machine where the job part is processed. During setup a machine cannot process or set up any other job. We concentrate on the basic case in which setup times are job-, machine-, and sequence-independent. Problems of this kind were encountered when modelling practical problems in planning disaster relief operations. Our main algorithmic result is a polynomial-time algorithm for minimising total completion time on two parallel identical machines. We argue why the same problem with three machines is not an easy extension of the two-machine case, leaving the complexity of this case as a tantalising open problem. We give a constant-factor approximation algorithm for the general case with any number of machines and a polynomial-time approximation scheme for a fixed number of machines. For the version with objective minimising weighted total completion time we prove NP-hardness. Finally, we conclude with an overview of the state of the art for other split scheduling problems with job-, machine-, and sequence-independent setup times.