Approximability of scheduling with fixed jobs

Abstract

Scheduling problems of minimizing the makespan on identical parallel machines are among the most wellstudied problems -especially in the ÿeld of approximation. In modern industrial software however, it has become standard to work on a variant of this problem, where some of the jobs are already ÿxed in the schedule. The remaining jobs are to be assigned to the machines in such a way that they do not overlap with ÿxed jobs. This problem variant is the root of many real-world scheduling problems where pre-assignments on the machines are considered, such as cleaning times or jobs that have already started. In our paper we investigate the approximability of the scheduling problem with ÿxed jobs. We present a polynomial-time approximation scheme (PTAS) for the case that the number m of machines is constant. For our PTAS we propose a new technique by partitioning an underlying packing problem into a reasonable unrelated family of restricted bin packing problems. We also generalize the PTAS to the case that the machines are independent and run at di erent speeds. Moreover, we will demonstrate that, assuming P = NP, there is no arbitrarily close approximation in the general case when the number of machines is part of the input. This will be extended by showing that there is no asymptotic PTAS in the general machine case. We ÿnally show that there exists no FPTAS in the constant machine case, unless P=NP. These results contrast to the classical problem of minimizing the makespan where the existence of a PTAS resp. of an FPTAS for the variable resp. the constant machine case has been proven.

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.…”

“…Scheduling with fixed jobs was studied by Scharbrodt, Steger & Weisser [28,29,30]. They mainly studied the problem for m constant; for this easier yet still strongly NP-hard formulation they present a PTAS.…”

“…They mainly studied the problem for m constant; for this easier yet still strongly NP-hard formulation they present a PTAS. They also found approximation algorithms for general m with ratios 3 [28] and 2 + [30]; since the finishing time of the last fixed job is a lower bound for C by Eyraud-Dubois, Mounié & Trystram [5]. The inapproximability is circumvented by requiring at least one machine to be permanently available.…”

Multi-agent scheduling on a single machine with max-form criteria

“…This scheduling problem was first introduced by Scharbrodt et al (1999). In this model we have two types of jobs: free jobs and fixed jobs.…”

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.…”