Advertisement

The Computational Complexity of the Game of Set and Its Theoretical Applications

  • Michael Lampis
  • Valia Mitsou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

The game of SET is a popular card game in which the objective is to form Sets using cards from a special deck. In this paper we study single- and multi-round variations of this game from the computational complexity point of view and establish interesting connections with other classical computational problems.

Specifically, we first show that a natural generalization of the problem of finding a single Set, parameterized by the size of the sought Set is W-hard; our reduction applies also to a natural parameterization of Perfect Multi-Dimensional Matching, a result which may be of independent interest. Second, we observe that a version of the game where one seeks to find the largest possible number of disjoint Sets from a given set of cards is a special case of 3-Set Packing; we establish that this restriction remains NP-complete. Similarly, the version where one seeks to find the smallest number of disjoint Sets that overlap all possible Sets is shown to be NP-complete, through a close connection to the Independent Edge Dominating Set problem. Finally, we study a 2- player version of the game, for which we show a close connection to Arc Kayles, as well as fixed-parameter tractability when parameterized by the number of rounds played.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrahamson, K.R., Downey, R.G., Fellows, M.R.: Fixed-Parameter Tractability and Completeness IV: On Completeness for W[P] and PSPACE Analogues. Ann. Pure Appl. Logic 73(3), 235–276 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chaudhuri, K., Godfrey, B., Ratajczak, D., Wee, H.: On the Complexity of the Game of Set (2003) (manuscript)Google Scholar
  3. 3.
    Chen, J., Feng, Q., Liu, Y., Lu, S., Wang, J.: Improved Deterministic Algorithms for Weighted Matching and Packing Problems. Theor. Comput. Sci. 412(23), 2503–2512 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Coleman, B., Hartshorn, K.: Game, Set, Math. Mathematics Magazine 85(2), 83–96 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Davis, B.L., Davis, Maclagan, D.: The Card Game Set (2003)Google Scholar
  6. 6.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the Parameterized Complexity of Multiple-interval Graph Problems. Theor. Comput. Sci. 410(1), 53–61 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 174. Freeman, New York (1979)zbMATHGoogle Scholar
  8. 8.
    Grier, D.: Deciding the Winner of an Arbitrary Finite Poset Game Is PSPACE-Complete. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 497–503. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Lampis, M., Mitsou, V.: The Computational Complexity of the Game of Set and its Theoretical Applications. arXiv preprint arXiv:1309.6504 (2013)Google Scholar
  10. 10.
    Papadimitriou, C.M.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  11. 11.
    Schaefer, T.J.: On the Complexity of Some Two-Person Perfect-Information Games. J. Comput. Syst. Sci. 16(2), 185–225 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. PhD thesis, Linköping (2007)Google Scholar
  13. 13.
    Zabrocki, M.: The joy of set (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Michael Lampis
    • 1
  • Valia Mitsou
    • 2
  1. 1.Research Institute for Mathematical Sciences (RIMS)Kyoto UniversityJapan
  2. 2.CUNY Graduate CenterUSA

Personalised recommendations