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. 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.

Improved Approximation Algorithms for Scheduling with Fixed Jobs

“…In parallel machine scheduling, an important issue is the scenario where either some jobs are already fixed in the system [29,30] or intervals of non-availability of some machines must be taken into account [4,11,21,23,24]. 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 [26] and overlay computing where machines are donated on a volunteer basis.…”

“…For the latter problem, we denote by ρ ∈ (0, 1) the percentage of machines which are permanently available and also permit infinite length of the non-availability intervals. In the literature, scheduling with non-availability is also called non-resumable scheduling with availability constraints [4,21,23,24]. The makespan C max is one of the most well-studied objectives in the field of scheduling and usually regarded as an "easy" objective in the sense that most problem formulations permit good approximation algorithms.…”

“…The inapproximability is circumvented by requiring at least one machine to be permanently available. The case with m constant, arbitrary non-availability intervals, and at least one machine permanently available, is strongly NPhard but can be solved by a PTAS by Diedrich et al [4]. For general m, researchers so far have only studied the problem where there is at most one interval of nonavailability per machine.…”

“…Concerning further results, we refer the reader to [23], Chapt. 22, or [27] for surveys and the articles [4,14,20] for results on singlemachine problems. For scheduling with non-availability, our new technique yields an improved approximation ratio independent from ρ which is basically tight.…”

Parameterized Approximation Scheme for the Multiple Knapsack Problem

“…The books on knapsack problems by Martello and Toth [15] and by Kellerer, Pferschy, and Pisinger [12] both have a full chapter devoted to MKP. An interesting application arises in scheduling jobs on identical processors with reservations or fixed jobs [5,18]. In this case either high-priority jobs are preassigned to machines or machines are non-available due to maintenance during fixed intervals.…”