Abstract
We consider the following shared-resource scheduling problem: Given a set of jobs J, for each \(j\in J\) we must schedule a job-specific processing volume of \(v_j>0\). A total resource of 1 is available at any time. Jobs have a resource requirement \(r_j\in \left[ 0,1 \right] \), and the resources assigned to them may vary over time. However, assigning them less will cause a proportional slowdown.
We consider two settings. In the first, we seek to minimize the makespan in an online setting: The resource assignment of a job must be fixed before the next job arrives. Here we give an optimal \(e/(e-1)\)-competitive algorithm with runtime \(\text {O}({n\text {\,\,} \log \, n})\). In the second, we aim to minimize the total completion time. We use a continuous linear programming (CLP) formulation for the fractional total completion time and combine it with a previously known dominance property from malleable job scheduling to obtain a lower bound on the total completion time. We extract structural properties by considering a geometrical representation of a CLP’s primal-dual pair. We combine the CLP schedule with a greedy schedule to obtain a \((3/2+\varepsilon )\)-approximation for this setting. This improves upon the so far best-known approximation factor of 2.
Full Version (Preprint): https://arxiv.org/abs/2310.05732.
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Notes
- 1.
One can think of \(U_V\) as a schedule for a single job of volume V and resource requirement 1. Since there is only one job, we identify \(U_V\) with its total resource requirement function \(\bar{U}_V\).
- 2.
- 3.
While not strictly required, this makes line schedules unique and simplifies the analysis.
- 4.
The second part of the definition (\(d_{j'}(t)=d_j(t)\) and \(v_{j'}>v_j\)) only exists for disambiguation of the line schedule, but is not further relevant.
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Damerius, C., Kling, P., Schneider, F. (2024). Improved Scheduling with a Shared Resource. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14461. Springer, Cham. https://doi.org/10.1007/978-3-031-49611-0_11
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