Empowering the Configuration-IP $-$ New PTAS Results for Scheduling with Setups 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 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/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. ACM Subject ClassificationTheory of computation → Scheduling algorithms, Theory of computation → Discrete optimization Keywords and phrases Parallel Machines, Setup Time, EPTAS, n-fold integer programming Acknowledgements We thank Syamantak Das for helpful discussions on the problem.

Block-structured Integer Programming: Can we Parameterize without the Largest Coefficient?

“…From now on we write d i (ξ h ) and di (ξ h ) as d i h and di h as they become fixed values. By Eq (15) we have…”

“…Such block-structured IP finds application in a variety of optimization problems including string matching, computational social choice, resource allocation, etc (see ,e.g. [23,9,24,3,15,22]). We give a brief introduction below.…”

New Bounds on Augmenting Steps of Block-structured Integer Programs

“…An extension of n-fold IP to tree-structured matrices called tree-fold IP was developed by Chen and Marx [5] and applied to scheduling problems. Jansen et al [19] have used n-fold IP to obtain efficient PTASes for scheduling problems. An extension of 2-stage stochastic IP analogous to tree-folds is called multi-stage stochastic and was studied by Aschenbrenner and Hemmecke [4].…”

“…Augmentation algorithms for IP and Graver-best oracle. There is a general framework for solving (IP) by utilizing G(A), which was developed in a series of papers [5,15,19,23]. A recent paper by Koutecký et al [25] formalizes this framework and extends it to also obtaining strongly polynomial algorithms (algorithms whose number of arithmetic operations does not depend on the length of the numbers on input).…”

“…Famous polynomially solvable cases are IPs with few rows and small coefficients as shown by Papadimitriou in 1981 [27], and IPs with few variables as shown by Lenstra in 1983. Arguably the most significant development in the last 20 years has been the introduction of iterative augmentation methods which led to the development of fast algorithms for wide classes of IPs whose constraint matrix has a special block structure, and to subsequent breakthrough applications in parameterized and approximation algorithms [5,19,23]. In fact, essentially all tractable classes of IP in variable dimension are of this kind, except for the theory of total unimodularity from the '60s.…”