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Polyamorous Scheduling

Authors Leszek Gąsieniec , Benjamin Smith , Sebastian Wild

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Scheduling algorithms
Keywords
  • Periodic scheduling
  • Pinwheel scheduling
  • Edge-coloring
  • Chromatic index
  • Approximation algorithms
  • Hardness of approximation

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Abstract

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, an O(log Δ)-approximation for Δ 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.

Cite As Get BibTex

Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild. Polyamorous Scheduling. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FUN.2024.15

Author Details

Leszek Gąsieniec
  • University of Liverpool, UK
Benjamin Smith
  • University of Liverpool, UK
Sebastian Wild
  • University of Liverpool, UK

Funding

  • Wild, Sebastian: Engineering and Physical Sciences Research Council grant EP/X039447/1.

Acknowledgements

We are grateful to Viktor Zamaraev for setting us on the right track with the chromatic-index problem, and for several fruitful initial discussions. We also wish to thank Casper Moldrup Rysgaard, Justin Dallant, and Oliver Kim for their helpful contributions; especially Casper, who acted as our rubber duck for a brutally unpolished version of the original hardness-of-approximation reduction.

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