Approximation Algorithms for Scheduling with Reservations

Abstract

We study the problem of non-preemptively scheduling n independent sequential jobs on a system of m identical parallel machines in the presence of reservations, where m is constant. This setting is practically relevant because for various reasons, some machines may not be available during specified time intervals. The objective is to minimize the makespan C max , which is the maximum completion time.An extended abstract of this work has been accepted at 392 Algorithmica (2010) 58: [391][392][393][394][395][396][397][398][399][400][401][402][403][404] The general case of the problem is inapproximable unless P = NP; hence, we study a suitable strongly NP-hard restriction, namely the case where at least one machine is always available. For this setting we contribute approximation schemes, complemented by inapproximability results. The approach is based on algorithms for multiple subset sum problems; our technique yields a PTAS which is best possible in the sense that an FPTAS is ruled out unless P = NP. The PTAS presented here is the first one for the problem under consideration; so far, not even for well-known special cases approximation schemes have been proposed. Furthermore we derive a low cost algorithm with a constant approximation ratio and discuss FPTASes for special cases as well as the complexity of the problem if m is part of the input.

Tight approximation algorithms for scheduling with fixed jobs and nonavailability

“…Concerning further results, we refer the reader to Leung [2004, Chapter 22], or Sanlaville and Schmidt [1998] for surveys. For the sake of completeness, some results about single-machine problems can be found in the articles [Diedrich et al 2010;Kacem 2009;Lee 1991]. Finally, for scheduling with nonavailability, our new technique yields an improved approximation ratio independent from ρ which is tight.…”

“…The inapproximability is circumvented by requiring at least one machine to be permanently available. The case with m constant, arbitrary nonavailability intervals, and at least one machine permanently available, is strongly NP-hard but Diedrich et al [2010] gave a PTAS for it. For general m, researchers so far have only studied the problem where there is at most one interval of nonavailability per machine.…”

“…In the literature, scheduling with nonavailability is also called nonresumable scheduling with availability constraints [Diedrich et al 2010;Lee 1996;Leung 2004;Liao et al 2003]. The makespan C max is one of the most well-studied objectives in the field of scheduling; for this objective, most problem formulations permit good approximation algorithms.…”

“…To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specific permission and/or a fee. Permissions may be requested from 27:2 F. Diedrich et al et al 2011], production planning and control for the beverage industry [Scharbrodt et al 1999], and distributed computing [Diedrich et al 2010].…”

“…In parallel machine scheduling, an important issue is the scenario where either some jobs are already fixed in the system [Scharbrodt et al 1999] or intervals of nonavailability of some machines must be taken into account [Diedrich et al 2010;Hwang et al 2005;Lee 1996;Leung 2004;Liao et al 2003]. The first problem occurs since high-priority jobs are present in the system while the latter problem is due to regular maintenance of machines; both models are relevant for turnaround scheduling [Megow This is a combined version of the work with the title "Improved Approximation Algorithms for Scheduling with Fixed Jobs" presented at the 20 th ACM-SIAM Symposium on Discrete Algorithms (SODA09) and the work with title "Faster Approximation Algorithms for Scheduling with Fixed Jobs" presented at the 17 th Computing: The Australian Theory Symposium (CATS11).…”