Empowering the configuration-IP: new PTAS results for scheduling with setup times

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 choose 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 $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 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.

Integer programs with bounded subdeterminants and two nonzeros per row

“…These methods can be found at the core of many approaches for combinatorial optimization problems. Illustrative examples are algorithms by Goemans and Rothvoß [16] and Jansen, Klein, Maack, and Rau [20] for bin packing and scheduling problems, respectively. One of the most fundamental efficiently solvable classes consists of integer programs of the form (IP) whose coefficient matrix A is totally unimodular, i.e., each square submatrix of A has a determinant within {−1, 0, 1}.…”

FPT Algorithms for a Special Block-structured Integer Program with Applications in Scheduling

“…In 2013, Hemmecke et al [20] developed the first FPT algorithm. Later on, a series of researches have been carried out to further improve its running time and also to extend the algorithm to the generalized n-fold IP [3,9,12,13,24,25]. Most recently, Cslovjecsek et al [9] presented an algorithm of running time 1) for the generalized n-fold IP.…”

“…Integer programs (IPs) whose constraint matrix has a special block structure have received a considerable attention in recent years. As an important subclass of the general IP, it finds applications in a variety of optimization problems including scheduling [7,24,28], routing [7], stochastic integer multi-commodity flows [18], stochastic programming with second-order dominance constraints [15], etc.…”

“…From a theoretical point of view, combinatorial 4-block n-fold IP exhibits an interesting "intermediate" phenomenon in its Graver basis (see Section 2 for the definition). As we will provide more details later in the related work, so far, FPT algorithms have been developed for several special cases of the generalized 4-block n-fold IP (see, e.g., [4,9,20,24,29]). All of these algorithms rely on the fact that the ℓ ∞ -norm (or even ℓ 1 -norm) of the Graver basis elements for these special cases are bounded by some FPT-value.…”

Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs

“…These ILPs often arise in the context of algorithmic applications. Probably the most famous one among such ILPs is the configuration ILP introduced by Gilmore and Gomory [10] and used for many packing and scheduling problems (e. g. [3,11,17,18,19]).…”