Abstract
Three men, each with a sister, must cross a river using a boat that can carry only two people in such a way that a sister is never left in the company of another man if her brother is not present. This very famous problem appeared in the Latin book “Problems to Sharpen the Young,” one of the earliest collections of recreational mathematics. This paper considers a generalization of such “river crossing problems” and provides a new formulation that can treat wide variations. The main result is that, if there is no upper bound on the number of transportations (river crossings), a large class of subproblems can be solved in polynomial time even when the passenger capacity of the boat is arbitrarily large. The authors speculated this fact at FUN 2012. On the other hand, this paper also demonstrates that, if an upper bound on the number of transportations is given, the problem is NP-hard even when the boat capacity is three, although a large class of subproblems can be solved in polynomial time if the boat capacity is two.
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Acknowledgments
We would like to express our gratitude to the anonymous referees for their truly careful reviews and very detailed comments. They significantly improved this article. We also thank JSPS KAKENHI Grant Number 24650006 and the ELC project (MEXT KAKENHI Grant Number 24106003), through which this work was partially supported.
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Ito, H., Langerman, S. & Yoshida, Y. Generalized River Crossing Problems. Theory Comput Syst 56, 418–435 (2015). https://doi.org/10.1007/s00224-014-9562-8
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DOI: https://doi.org/10.1007/s00224-014-9562-8