Abstract
We consider a scheduling problem with resource-dependent processing speeds in which n jobs have to be scheduled on m machines that share a common resource. The resource may be distributed arbitrarily among the machines. This distribution is under the control of the scheduler and can be changed over time. Each job j has a processing volume \(p_j \in \mathbb {N}\) and a resource requirement \(r_j \in (0,1]\). The latter indicates what fraction of the resource a job requires to run at full speed. Providing it with a larger share is not beneficial, but lowering its share results in a proportionally lowered processing speed. The goal is to schedule all jobs non-preemptively while minimizing the latest completion time.
This problem was introduced by Kling et al. [SPAA’17], who proved NP-hardness and gave an efficient algorithm with approximation ratio \(2+1/(m-2)\). The (asymptotic) tightness of that bound was left as an open question. We focus on the case of two machines and derive a strong, structural lower bound. This lower bound is based on a relaxed version and allows us to design an asymptotic 3/2-approximation that runs in time \({\text {O}}\left( n \cdot \log n\right) \). As an immediate consequence we also get an improved 9/4-approximation for the case of three machines.
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Notes
- 1.
In [10], \(r_j > 1\) is allowed. Our restriction is without loss of generality, as we can assign such jobs a resource requirement of 1 and increase their processing volume by a factor \(r_j\) to get an equivalent instance.
- 2.
Alternatively, one can allow resource requirements \(>1\) and use these as item sizes while setting all processing volumes to 1, as described in [10].
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Acknowledgement
Peter Kling and Christoph Damerius were partially supported by the DAAD PPP with Project-ID 57447553. Minming Li is also from City University of Hong Kong Shenzhen Research Institute, Shenzhen, P.R. China. The work described in this paper was partially supported by Project 11771365 supported by NSFC.
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Damerius, C., Kling, P., Li, M., Schneider, F., Zhang, R. (2020). Improved Scheduling with a Shared Resource via Structural Insights. In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_12
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