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

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. Configuration IPs are especially useful for problems in which the items and target locations are rather homogeneous, as is the case in the classical problems of scheduling on identical machines and bin packing. However, when considering, for instance, problems in which the items are partitioned into classes and have some class dependent property, the configuration-IP quickly encounters its limits. We give an augmented IP formulation, which we call the module configuration IP. This IP cannot be solved efficiently using the classical algorithm of Lenstra and Kannan. Instead, we show that it can be described within the framework of n-fold integer programming and therefore be solved efficiently using the algorithm by Hemmecke, Onn, and Romanchuk. Furthermore, we investigate how structural properties can be applied to reduce the description of the IP and thereby speed up the resulting algorithms. 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/ε)× 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 + ε 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. This work was partially supported by German Research Foundation (DFG) project JA 612/20-1