Empowering the Configuration-IP $-$ New PTAS Results for Scheduling with Setups Times

TL;DR

This paper introduces a new module configuration integer linear programming approach, enabling efficient approximation schemes for complex scheduling problems with setup times, significantly improving over previous constant factor approximations.

Contribution

The authors develop an augmented configuration IP within n-fold integer programming, providing the first EPTAS for scheduling with setup times, including parallel and non-splittable job variants.

Findings

EPTAS with single exponential time for parallel jobs

Double exponential time EPTAS for non-parallel jobs

Improved approximation ratios from previous constant factors

Abstract

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems, where a set of items has to be placed in multiple target locations. Herein a configuration describes a possible placement on one of the target locations, and the IP is used to chose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and therefore be solved efficiently. As an application, we consider scheduling problems with setup times, in which a set of jobs has to be scheduled on a set of identical machines, with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $f(1/\varepsilon)\times \mathrm{poly}(|I|)$ with a single exponential term in $f$ for the first and a double exponential one for the second case. Previously, only constant factor approximations of $5/3$ and $4/3 + \varepsilon$ respectively were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.