Approximation Schemes for Machine Scheduling with Resource (In-)dependent Processing Times

Abstract

We consider two related scheduling problems: single resource-constrained scheduling on identical parallel machines and a generalization with resource-dependent processing times. In both problems, jobs require a certain amount of an additional resource and have to be scheduled on machines minimizing the makespan, while at every point in time a given resource capacity is not exceeded. In the first variant of the problem, the processing times and resource amounts are fixed, while in the second the former depends on the latter. Both problems contain bin packing with cardinality constraint as a special case, and, therefore, these problems are strongly NP-complete even for a constant number of machines larger than three, which can be proven by a reduction from 3-Partition. Furthermore, if the number of machines is part of the input, then we cannot hope for an approximation algorithm with absolute approximation ratio smaller than 3/2. We present asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems: For any ε > 0, a schedule of length at most (1+ε) times the optimum plus an additive term of O ( p max log (1/ε)/ε) is provided, and the running time is polynomially bounded in 1/ε and the input length. Up to now, only approximation algorithms with absolute approximation ratios were known. Furthermore, the AFPTAS for resource-constrained scheduling on identical parallel machines directly improves the additive term of the best AFPTAS for bin packing with cardinality constraint so far.

A Log-Linear $$(2 +5/6)$$-Approximation Algorithm for Parallel Machine Scheduling with a Single Orthogonal Resource

“…Our algorithm improves the 3 − 3 m -approximation ratio of Garey and Graham [6]. It is also considerably easier to implement than the approximation algorithms proposed by Niemeier and Wiese [21] and Jansen, Maack and Rau [14].…”

“…This algorithm can be transformed into a PTAS if the number of machines or the number of different resource requirements is upper-bounded by a constant. Further, an asymptotic FPTAS for the P|res1••|C max problem was shown [14]. Although the latter results have a great theoretical impact on the problem considered in this paper, the complexity of the obtained algorithms prevents them from being efficiently implemented.…”

“…We thus directly improve the classic 3 − 3 m -approximation algorithm by Garey and Graham [6]. Although a (2 + ε)-approximation algorithm [21] and an asymptotic FPTAS [14] are known, their time and implementation complexities are considerable. Our algorithm can be easily implemented and it runs in log-linear time.…”

Closing the gap for single resource constraint scheduling

“…While for the placement of small jobs, we use techniques already known, see e.g. [24], we used a new technique to place the large jobs. For the following (3/2 + ε) approximation, it is important to guarantee that only O ε (1), i.e.…”

“…For bin packing with cardinality constraint Epstein and Levin [10] presented an AFPTAS. This AFPTAS was improved and extended to work for single resource constraint scheduling by Jansen et al [24]. It has an additive term of O(p max log(1/ε)/ε).…”