Polyamorous Scheduling
<|reference_start|>Polyamorous Scheduling: Finding schedules for pairwise meetings between the members of a complex social group without creating interpersonal conflict is challenging, especially when different relationships have different needs. We formally define and study the underlying optimisation problem: Polyamorous Scheduling.   In Polyamorous Scheduling, we are given an edge-weighted graph and try to find a periodic schedule of matchings in this graph such that the maximal weighted waiting time between consecutive occurrences of the same edge is minimised. We show that the problem is NP-hard and that there is no efficient approximation algorithm with a better ratio than 4/3 unless P = NP. On the positive side, we obtain an $O(\log n)$-approximation algorithm; indeed, a $O(\log \Delta)$-approximation for $\Delta$ the maximum degree, i.e., the largest number of relationships of any individual. We also define a generalisation of density from the Pinwheel Scheduling Problem, "poly density", and ask whether there exists a poly-density threshold similar to the 5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024]. Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with respect to its optimisation variant, Bamboo Garden Trimming.   Our work contributes the first nontrivial hardness-of-approximation reduction for any periodic scheduling problem, and opens up numerous avenues for further study of Polyamorous Scheduling.<|reference_end|>
@article{gąsieniec2024polyamorous,
    title={Polyamorous Scheduling},
    author={Leszek Gk{a}sieniec and Benjamin Smith and Sebastian Wild},
    journal={arXiv preprint arXiv:2403.00465},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.00465},
    primaryClass={cs.DS cs.SI math.OC}
}