Algorithms and Complexity of Generalized River Crossing Problems

  • Hiro Ito
  • Stefan Langerman
  • Yuichi Yoshida
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7288)

Abstract

Three men, each with a sister, must cross a river using a boat which can carry only two people, so that a woman whose brother is not present is never left in the company of another man. This is a very famous problem appeared in Latin book “Problems to Sharpen the Young,” one of the earliest collections on recreational mathematics. This paper considers a generalization of such “River-Crossing Problems.” It shows that the problem is NP-hard if the boat size is three, and a large class of sub-problems can be solved in polynomial time if the boat size is two. It’s also conjectured that determining whether a river crossing problem has a solution without bounding the number of transportations, can be solved in polynomial time even when the size of the boat is large.

Keywords

Polynomial Time Vertex Cover Left Bank River Crossing Famous Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahls, P.: The wolf, the goat, and the cabbage: A modern twist on a classical problem, http://facstaff.unca.edu/pbahls/talks/WGC.pdf
  2. 2.
    Bellman, R.: Dynamic programming and “difficult crossing” puzzles. Mathematics Magazine 35(1), 27–29 (1962)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Borndörfer, R., Grötschel, M., Löbel, A.: Alcuin’s transportation problems and integer programming, Preprint SC-95-27, Konrad-Zuse-Zentrum für Informationstechnik Berlin (1995)Google Scholar
  4. 4.
    Csorba, P., Hurkens, C.A.J., Woeginger, G.J.: The Alcuin Number of a Graph. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 320–331. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Csorba, P., Hurkens, C.A.J., Woeginger, G.J.: The Alcuin number of a graph and its connections to the vertex cover number. SIAM J. Discrete Math. 24(3), 757–769 (2010)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman And Company (1979)Google Scholar
  7. 7.
  8. 8.
    Lampis, M., Mitsou, V.: The Ferry Cover Problem. In: Crescenzi, P., Prencipe, G., Pucci, G. (eds.) FUN 2007. LNCS, vol. 4475, pp. 227–239. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Peterson, I.: Tricky crossings, Science News Online 164(24), http://web.archive.org/web/20040603203306/http://www.sciencenews.org/articles/20031213/mathtrek.asp (retrieved February 7, 2008)
  10. 10.
    Schwartz, B.R.: An analytic method for the “difficult crossing” puzzles. Mathematics Magazine 34(4), 187–193 (1961)MATHCrossRefGoogle Scholar
  11. 11.
    Trevisan, L.: Graph Partitioning and Expanders, Stanford University — CS359G, Lecture 6 (2011), http://theory.stanford.edu/~trevisan/cs359g/
  12. 12.
    Propositiones ad Acuendos Juvenes, Wikipedia, the free encyclopediaGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hiro Ito
    • 1
  • Stefan Langerman
    • 2
  • Yuichi Yoshida
    • 1
    • 3
  1. 1.School of InformaticsKyoto UniversityKyotoJapan
  2. 2.Maître de recherches du F.R.S.-FNRS, Département d’informatiqueUniversité Libre de Bruxelles (ULB)Belgium
  3. 3.Preferred Infrastructure, Inc.Japan

Personalised recommendations