Scheduling jobs with given processing times on identical parallel machines so as to minimize their total completion time is one of the most basic scheduling problems. We study interesting generalizations of this classical problem involving scenarios. In our model, a scenario is defined as a subset of a predefined and fully specified set of jobs. The aim is to find an assignment of the whole set of jobs to identical parallel machines such that the schedule, obtained for the given scenarios by simply skipping the jobs not in the scenario, optimizes a function of the total completion times over all scenarios. While the underlying scheduling problem without scenarios can be solved efficiently by a simple greedy procedure (SPT rule), scenarios, in general, make the problem NP-hard. We paint an almost complete picture of the evolving complexity landscape, drawing the line between easy and hard. One of our main algorithmic contributions relies on a deep structural result on the maximum imbalance of an optimal schedule, based on a subtle connection to Hilbert bases of a related convex cone.

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Lecture Notes in Computer Science
Optimization for and with Machine Learning , Networks
21st International Workshop on Approximation and Online Algorithms, WAOA 2023
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Bosman, T., van Ee, M., Ergen, E., Imreh, C., Marchetti Spaccamela, A., Skutella, M. (Martin), & Stougie, L. (2023). Total completion time scheduling under scenarios. In Proceedings of the 21st International Workshop on Approximation and Online Algorithms, WAOA 2023 (pp. 104–118). doi:10.1007/978-3-031-49815-2_8