Document Open Access Logo

A Tractability Gap Beyond Nim-Sums: It’s Hard to Tell Whether a Bunch of Superstars Are Losers

Authors Kyle Burke , Matthew Ferland , Svenja Huntemann , Shanghua Teng

PDF
Thumbnail PDF

File

LIPIcs.FUN.2024.8.pdf
  • Filesize: 0.88 MB
  • 14 pages

Document Identifiers

Author Details

Kyle Burke
  • Florida Southern College, Lakeland, FL, USA
Matthew Ferland
  • University of Southern California, Los Angeles, CA, USA
Svenja Huntemann
  • Mount Saint Vincent University, Halifax, Canada
Shanghua Teng
  • University of Southern California, Los Angeles, CA, USA

Funding

  • Teng, Shanghua: Supported in part by NSF Grant CCF-2308744 and the Simons Investigator Award from the Simons Foundation.

Cite AsGet BibTex

Kyle Burke, Matthew Ferland, Svenja Huntemann, and Shanghua Teng. A Tractability Gap Beyond Nim-Sums: It’s Hard to Tell Whether a Bunch of Superstars Are Losers. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FUN.2024.8

Abstract

In this paper, we address a natural question at the intersection of combinatorial game theory and computational complexity: "Can a sum of simple tepid games in canonical form be intractable?" To resolve this fundamental question, we consider superstars, positions first introduced in Winning Ways where all options are nimbers. Extending Morris' classic result with hot games to tepid games, we prove that disjunctive sums of superstars are intractable to solve. This is striking as sums of nimbers can be computed in linear time. Our analyses also lead to a family of elegant board games with intriguing complexity, for which we present web-playable versions of the rulesets described within.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Combinatorial Game Theory
  • NP-hardness
  • Superstars

Metrics

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

References

  1. M. H. Albert, R. J. Nowakowski, and D. Wolfe. Lessons in Play: An Introduction to Combinatorial Game Theory. A. K. Peters, Wellesley, Massachusetts, 2007. Google Scholar
  2. Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. Winning Ways for your Mathematical Plays, volume 1. A K Peters, Wellesley, Massachsetts, 2001. Google Scholar
  3. Charles L. Bouton. Nim, a game with a complete mathematical theory. Annals of Mathematics, 3(1/4):pp. 35-39, 1901. URL: http://www.jstor.org/stable/1967631.
  4. Kyle Burke, Matthew Ferland, and Shang-Hua Teng. Winning the war by (strategically) losing battles: Settling the complexity of Grundy-values in undirected geography. In Proceedings of the 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2021. Google Scholar
  5. J.H. Conway. On Numbers and Games. A.K. Peters. A.K. Peters, 2000. URL: https://books.google.com/books?id=tXiVo8qA5PQC.
  6. Aviezri S. Fraenkel, Edward R. Scheinerman, and Daniel Ullman. Undirected edge geography. Theor. Comput. Sci., 112(2):371-381, 1993. Google Scholar
  7. M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified np-complete problems. In Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, STOC '74, pages 47-63, New York, NY, USA, 1974. Association for Computing Machinery. URL: https://doi.org/10.1145/800119.803884.
  8. P. M. Grundy. Mathematics and games. Eureka, 2:198 - -211, 1939. Google Scholar
  9. David Moews. On some combinatorial games connected with Go. PhD thesis, Citeseer, 1993. Google Scholar
  10. FL Morris. Playing disjunctive sums is polynomial space complete. International Journal of Game Theory, 10(3-4):195-205, 1981. Google Scholar
  11. Thomas J. Schaefer. The complexity of satisfiability problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC '78, pages 216-226, New York, NY, USA, 1978. Association for Computing Machinery. URL: https://doi.org/10.1145/800133.804350.
  12. A.N. Siegel. Combinatorial Game Theory. Graduate Studies in Mathematics. American Mathematical Society, 2013. URL: https://books.google.com/books?id=VUVrAAAAQBAJ.
  13. Alexandre Silva, Carlos Pereira dos Santos, João Pedro Neto, and Richard J. Nowakowski. Disjunctive sums of quasi-nimbers. Theoretical Computer Science, 945:113665, 2023. URL: https://doi.org/10.1016/j.tcs.2022.12.015.
  14. R. P. Sprague. Über mathematische Kampfspiele. Tôhoku Mathematical Journal, 41:438 - -444, 1935-36. Google Scholar
  15. David Wolfe. Go endgames are PSPACE-hard. In Richard J. Nowakowski, editor, More Games of No Chance, volume 42 of Mathematical Sciences Research Institute Publications, pages 125-136. Cambridge University Press, 2002. Google Scholar
  16. Laura Jo Yedwab. On playing well in a sum of games. PhD thesis, Massachusetts Institute of Technology, 1985. Google Scholar
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact