Document Open Access Logo

Swapping Mixed-Up Beers to Keep Them Cool

Authors Davide Bilò , Maurizio Fiusco, Luciano Gualà , Stefano Leucci

PDF
Thumbnail PDF

File

LIPIcs.FUN.2024.5.pdf
  • Filesize: 1.09 MB
  • 18 pages

Document Identifiers

Author Details

Davide Bilò
  • Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Italy
Maurizio Fiusco
  • Department of Enterprise Engineering, University of Rome "Tor Vergata", Italy
Luciano Gualà
  • Department of Enterprise Engineering, University of Rome "Tor Vergata", Italy
Stefano Leucci
  • Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Italy

Cite AsGet BibTex

Davide Bilò, Maurizio Fiusco, Luciano Gualà, and Stefano Leucci. Swapping Mixed-Up Beers to Keep Them Cool. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FUN.2024.5

Abstract

There was a mix-up in Escher’s bar and n customers sitting at the same table have each received a beer ordered by somebody else in the party. The drinks can be rearranged by swapping them in pairs, but the eccentric table shape only allows drinks to be exchanged between people sitting on opposite sides of the table. We study the problem of finding the minimum number of swaps needed so that each customer receives its desired beer before it gets warm. Formally, we consider the Colored Token Swapping problem on complete bipartite graphs. This problem is known to be solvable in polynomial time when all ordered drinks are different [Yamanaka et al., FUN 2014], but no results are known for the more general case in which multiple people in the party can order the same beer. We prove that Colored Token Swapping on complete bipartite graphs is NP-hard and that it is fixed-parameter tractable when parameterized by the number of distinct types of beer served by the bar.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Colored Token Swapping
  • Complete Bipartite Graphs
  • Labeled Token Swapping
  • FPT Algorithms
  • NP-Hardness

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Oswin Aichholzer, Erik D. Demaine, Matias Korman, Anna Lubiw, Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, and Nicole Wein. Hardness of token swapping on trees. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 3:1-3:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.3.
  2. Ahmad Biniaz, Kshitij Jain, Anna Lubiw, Zuzana Masárová, Tillmann Miltzow, Debajyoti Mondal, Anurag Murty Naredla, Josef Tkadlec, and Alexi Turcotte. Token swapping on trees. Discret. Math. Theor. Comput. Sci., 24(2), 2022. URL: https://doi.org/10.46298/DMTCS.8383.
  3. Édouard Bonnet, Tillmann Miltzow, and Pawel Rzazewski. Complexity of token swapping and its variants. Algorithmica, 80(9):2656-2682, 2018. URL: https://doi.org/10.1007/S00453-017-0387-0.
  4. Arthur Cayley. Lxxvii. note on the theory of permutations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(232):527-529, 1849. Google Scholar
  5. Matt Groening, James L. Brooks, Sam Simon, Dan Castellaneta, and Deb Lacusta. The Simpsons. Fox Broadcasting Company, 1999. Season 11, Episode 18 (Days of Wine and D'oh’ses), Timestamp: 18:30. URL: https://www.imdb.com/title/tt0701222.
  6. Lenwood S. Heath and John Paul C. Vergara. Sorting by short swaps. J. Comput. Biol., 10(5):775-789, 2003. URL: https://doi.org/10.1089/106652703322539097.
  7. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/JCSS.2000.1727.
  8. Mark Jerrum. The complexity of finding minimum-length generator sequences. Theor. Comput. Sci., 36:265-289, 1985. URL: https://doi.org/10.1016/0304-3975(85)90047-7.
  9. Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Math. Oper. Res., 8(4):538-548, 1983. URL: https://doi.org/10.1287/MOOR.8.4.538.
  10. Jun Kawahara, Toshiki Saitoh, and Ryo Yoshinaka. The time complexity of permutation routing via matching, token swapping and a variant. J. Graph Algorithms Appl., 23(1):29-70, 2019. URL: https://doi.org/10.7155/JGAA.00483.
  11. Tillmann Miltzow, Lothar Narins, Yoshio Okamoto, Günter Rote, Antonis Thomas, and Takeaki Uno. Approximation and hardness of token swapping. In Piotr Sankowski and Christos D. Zaroliagis, editors, 24th Annual European Symposium on Algorithms, ESA 2016, August 22-24, 2016, Aarhus, Denmark, volume 57 of LIPIcs, pages 66:1-66:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.ESA.2016.66.
  12. Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 974-988. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
  13. Theresa P. Vaughan. Factoring a permutation on a broom. Journal of Combinatorial Mathematics and Combinatorial Computing, 30:129-148, 1999. Google Scholar
  14. Katsuhisa Yamanaka, Erik D. Demaine, Takehiro Ito, Jun Kawahara, Masashi Kiyomi, Yoshio Okamoto, Toshiki Saitoh, Akira Suzuki, Kei Uchizawa, and Takeaki Uno. Swapping labeled tokens on graphs. Theor. Comput. Sci., 586:81-94, 2015. URL: https://doi.org/10.1016/J.TCS.2015.01.052.
  15. Katsuhisa Yamanaka, Takashi Horiyama, J. Mark Keil, David G. Kirkpatrick, Yota Otachi, Toshiki Saitoh, Ryuhei Uehara, and Yushi Uno. Swapping colored tokens on graphs. Theor. Comput. Sci., 729:1-10, 2018. URL: https://doi.org/10.1016/J.TCS.2018.03.016.
  16. Katsuhisa Yamanaka, Takashi Horiyama, David G. Kirkpatrick, Yota Otachi, Toshiki Saitoh, Ryuhei Uehara, and Yushi Uno. Swapping colored tokens on graphs. In Frank Dehne, Jörg-Rüdiger Sack, and Ulrike Stege, editors, Algorithms and Data Structures - 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015. Proceedings, volume 9214 of Lecture Notes in Computer Science, pages 619-628. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21840-3_51.
  17. Gaku Yasui, Kouta Abe, Katsuhisa Yamanaka, and Takashi Hirayama. Swapping labeled tokens on complete split graphs. Inf. Process. Soc. Japan. SIG Tech. Rep, 2015(14):1-4, 2015. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact