The Complexity of Escaping Labyrinths and Enchanted Forests
The board games The aMAZEing Labyrinth (or simply Labyrinth for short) and Enchanted Forest published by Ravensburger are seemingly simple family games.
In Labyrinth, the players move though a labyrinth in order to collect specific items. To do so, they shift the tiles making up the labyrinth in order to open up new paths (and, at the same time, close paths for their opponents). We show that, even without any opponents, determining a shortest path (i.e., a path using the minimum possible number of turns) to the next desired item in the labyrinth is strongly NP-hard. Moreover, we show that, when competing with another player, deciding whether there exists a strategy that guarantees to reach one's next item faster than one's opponent is PSPACE-hard.
In Enchanted Forest, items are hidden under specific trees and the objective of the players is to report their locations to the king in his castle. Movements are performed by rolling two dice, resulting in two numbers of fields one has to move, where each of the two movements must be executed consecutively in one direction (but the player can choose the order in which the two movements are performed). Here, we provide an efficient polynomial-time algorithm for computing a shortest path between two fields on the board for a given sequence of die rolls, which also has implications for the complexity of problems the players face in the game when future die rolls are unknown.
board games
combinatorial game theory
computational complexity
Theory of computation~Shortest paths
Theory of computation~Representations of games and their complexity
30:1-30:13
Regular Paper
Florian D.
Schwahn
Florian D. Schwahn
Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 14, D-67663 Kaiserslautern, Germany
The work of this author was partially supported by Anne M. Schwahn by explaining the two studied board games to him back in the dark winters of 1990 and 1991.
Clemens
Thielen
Clemens Thielen
Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 14, D-67663 Kaiserslautern, Germany
https://orcid.org/0000-0003-0897-3571
10.4230/LIPIcs.FUN.2018.30
E. D. Demaine and R. A. Hearn. Playing games with algorithms: Algorithmic combinatorial game theory, 2001. URL: http://arxiv.org/abs/cs.CC/0106019.
http://arxiv.org/abs/cs.CC/0106019
C. Dodaro, M. Alviano, W. Faber, N. Leone, F. Ricca, and M. Sirianni. The birth of a WASP: Preliminary report on a new ASP solver. In Proceedings of the 26th Italian Conference on Computational Logic (CILC), pages 99-113, 2011.
D. Eppstein. Computational complexity of games and puzzles. https://www.ics.uci.edu/~eppstein/cgt/hard.html. Accessed: 2018-04-16.
https://www.ics.uci.edu/~eppstein/cgt/hard.html
M. R. Garey and D. S. Johnson. Computers and Intractability (A Guide to the Theory of NP-Completeness). W.H. Freeman and Company, New York, 1979.
G. Kendall, A. J. Parkes, and K. Spoerer. A survey of NP-complete puzzles. ICGA Journal, 31(1):13-34, 2008.
M. J. Kobbert. Das verrückte Labyrinth. Ravensburger, 1986.
F. Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation (ISSAC), pages 296-303, 2014.
M. Matschoss and A. Randolph. Sagaland. Ravensburger, 1981.
C. Papadimitriou. Computational Complexity. Addison Wesley, 1993.
Spiel des Jahres e.V. Spiel des Jahres. URL: http://www.spiel-des-jahres.com/en.
http://www.spiel-des-jahres.com/en
R. P. Stanley. Enumerative Combinatorics. Wadsworth Publ. Co., 1986.
L. J. Stockmeyer and A. K. Chandra. Provably difficult combinatorial games. SIAM Journal on Computing, 8(2):151-174, 1979.
Florian D. Schwahn and Clemens Thielen
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode