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Polyamorous Scheduling

Authors Leszek Gąsieniec , Benjamin Smith , Sebastian Wild

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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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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

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.

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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References

  1. Peyman Afshani, Mark de Berg, Kevin Buchin, Jie Gao, Maarten Löffler, Amir Nayyeri, Benjamin Raichel, Rik Sarkar, Haotian Wang, and Hao-Tsung Yang. On cyclic solutions to the min-max latency multi-robot patrolling problem. In International Symposium on Computational Geometry (SoCG). Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.SOCG.2022.2.
  2. Amotz Bar-Noy, Richard E Ladner, and Tami Tamir. Windows scheduling as a restricted version of bin packing. ACM Transactions on Algorithms, 3(3):28-es, 2007. URL: https://doi.org/10.1145/1273340.1273344.
  3. Amotz Bar-Noy, Joseph (Seffi) Naor, and Baruch Schieber. Pushing dependent data in clients-providers-servers systems. Wireless Networks, 9(5):421-430, 2003. URL: https://doi.org/10.1023/a:1024632031440.
  4. Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Giacomo Scornavacca. Cutting Bamboo down to Size. In Martin Farach-Colton, Giuseppe Prencipe, and Ryuhei Uehara, editors, Fun with Algorithms (FUN), volume 157 of Leibniz International Proceedings in Informatics (LIPIcs), pages 5:1-5:18, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.FUN.2021.5.
  5. Thomas Bosman, Martijn van Ee, Yang Jiao, Alberto Marchetti-Spaccamela, R. Ravi, and Leen Stougie. Approximation algorithms for replenishment problems with fixed turnover times. Algorithmica, 84(9):2597-2621, May 2022. URL: https://doi.org/10.1007/s00453-022-00974-4.
  6. Mee Yee Chan and Francis Chin. Schedulers for larger classes of pinwheel instances. Algorithmica, 9(5):425-462, 1993. URL: https://doi.org/10.1007/BF01187034.
  7. Mee Yee Chan and Francis Y. L. Chin. General schedulers for the pinwheel problem based on double-integer reduction. IEEE Trans. Computers, 41(6):755-768, 1992. URL: https://doi.org/10.1109/12.144627.
  8. Wun-Tat Chan and Prudence W. H. Wong. On-line windows scheduling of temporary items. In International Symposium on Algorithms and Computation (ISAAC), pages 259-270. Springer Berlin Heidelberg, 2004. URL: https://doi.org/10.1007/978-3-540-30551-4_24.
  9. Serafino Cicerone, Gabriele Di Stefano, Leszek Gasieniec, Tomasz Jurdzinski, Alfredo Navarra, Tomasz Radzik, and Grzegorz Stachowiak. Fair hitting sequence problem: Scheduling activities with varied frequency requirements. In International Conference on Algorithms and Complexity (CIAC), pages 174-186. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-17402-6_15.
  10. Larry Clemmons, Ken Anderson, and Vance Gerry. Robin hood (movie). Walt Disney Productions, 1973. Google Scholar
  11. Wei Ding. A branch-and-cut approach to examining the maximum density guarantee for pinwheel schedulability of low-dimensional vectors. Real-Time Systems, 56(3):293-314, 2020. URL: https://doi.org/10.1007/s11241-020-09349-w.
  12. Eugene A. Feinberg and Michael T. Curry. Generalized pinwheel problem. Math. Methods Oper. Res., 62(1):99-122, 2005. URL: https://doi.org/10.1007/s00186-005-0443-4.
  13. Peter C Fishburn and Jeffrey C Lagarias. Pinwheel scheduling: Achievable densities. Algorithmica, 34(1):14-38, 2002. URL: https://doi.org/10.1007/s00453-002-0938-9.
  14. Leszek Gąsieniec, Benjamin Smith, and Sebastian Wild. Towards the 5/6-density conjecture of pinwheel scheduling. In C. A. Phillips and B. Speckmann, editors, Symposium on Algorithm Engineering and Experiments (ALENEX), pages 91-103. SIAM, January 2022. URL: https://doi.org/10.1137/1.9781611977042.8.
  15. Leszek Gąsieniec, Tomasz Jurdziński, Ralf Klasing, Christos Levcopoulos, Andrzej Lingas, Jie Min, and Tomasz Radzik. Perpetual maintenance of machines with different urgency requirements. Journal of Computer and System Sciences, 139:103476, February 2024. URL: https://doi.org/10.1016/j.jcss.2023.103476.
  16. Leszek Gąsieniec, Ralf Klasing, Christos Levcopoulos, Andrzej Lingas, Min Jie, and Tomasz Radzik. Bamboo Garden Trimming Problem, volume 10139 of Lecture Notes in Computer Science. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-51963-0.
  17. C.-C. Han and K.-J. Lin. Scheduling distance-constrained real-time tasks. In Proceedings Real-Time Systems Symposium. IEEE Comput. Soc. Press, 1992. URL: https://doi.org/10.1109/REAL.1992.242649.
  18. Felix Höhne and Rob van Stee. A 10/7-approximation for discrete bamboo garden trimming and continuous trimming on star graphs. In Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.16.
  19. Robert Holte, Al Mok, Al Rosier, Igor Tulchinsky, and Igor Varvel. The pinwheel: a real-time scheduling problem. In Proceedings of the Twenty-Second Annual Hawaii International Conference on System Sciences. Volume II: Software Track, volume 2, pages 693-702 vol.2, 1989. URL: https://doi.org/10.1109/HICSS.1989.48075.
  20. Robert Holte, Louis E. Rosier, Igor Tulchinsky, and Donald A. Varvel. Pinwheel scheduling with two distinct numbers. Theor. Comput. Sci., 100(1):105-135, 1992. URL: https://doi.org/10.1016/0304-3975(92)90365-M.
  21. Ian Holyer. The np-completeness of edge-coloring. SIAM Journal on computing, 10(4):718-720, 1981. Google Scholar
  22. Tobias Jacobs and Salvatore Longo. A new perspective on the windows scheduling problem. coRR, 2014. URL: https://arxiv.org/abs/1410.7237.
  23. John_Threepwood. Why not both? / Why don’t we have both?, August 2011. URL: https://knowyourmeme.com/memes/why-not-both-why-dont-we-have-both.
  24. Akitoshi Kawamura. Proof of the density threshold conjecture for pinwheel scheduling. In Symposium on Theory of Computing (STOC), 2024. URL: https://www.kurims.kyoto-u.ac.jp/~kawamura/pinwheel/paper_e.pdf.
  25. Akitoshi Kawamura and Makoto Soejima. Simple strategies versus optimal schedules in multi-agent patrolling. Theoretical Computer Science, 839:195-206, November 2020. URL: https://doi.org/10.1016/j.tcs.2020.07.037.
  26. Shun-Shii Lin and Kwei-Jay Lin. A pinwheel scheduler for three distinct numbers with a tight schedulability bound. Algorithmica, 19(4):411-426, 1997. URL: https://doi.org/10.1007/PL00009181.
  27. Jayadev Misra and David Gries. A constructive proof of vizing’s theorem. In Information Processing Letters. Citeseer, 1992. Google Scholar
  28. Martijn van Ee. A 12/7-approximation algorithm for the discrete bamboo garden trimming problem. Operations Research Letters, 49(5):645-649, September 2021. URL: https://doi.org/10.1016/j.orl.2021.07.001.
  29. Vadim G Vizing. The chromatic class of a multigraph. Cybernetics, 1(3):32-41, 1965. Google Scholar
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