Computational Complexity of Two-Dimensional Platform Games
We analyze the computational complexity of various two-dimensional platform games. We state and prove several meta-theorems that identify a class of these games for which the set of solvable levels is NP-hard, and another class for which the set is even PSPACE-hard. Notably CommanderKeen is shown to be NP-hard, and PrinceOfPersia is shown to be PSPACE-complete.
We then analyze the related game Lemmings, where we construct a set of instances which only have exponentially long solutions. This shows that an assumption by Cormode in  is false and invalidates the proof that the general version of the Lemmings decision problem is in NP. We then augment our construction to only include one entrance, which makes our instances perfectly natural within the context of the original game.
Unable to display preview. Download preview PDF.
- 1.Bouton, C.L.: Nim, a game with a complete mathematical theory. Annals of Mathematics 3, 35–39 (1901/2002)Google Scholar
- 3.Cormode, G.: The Hardness of the Lemmings Game, or Oh no, more NP-Completeness Proofs. In: Proceedings of Third International Conference on Fun with Algorithms, pp. 65–76 (2004)Google Scholar
- 4.Culbertson, J.: Sokoban is PSPACE-complete. In: Proceedings of the International Conference on Fun with Algorithms, pp. 65–76 (1998)Google Scholar
- 5.Demaine, E.D., Hearn, R.A.: Constraint logic: A uniform framework for modeling computation as games. In: Proceedings of the 23rd Annual IEEE Conference on Computational Complexity (2008)Google Scholar
- 6.Demaine, E.D., Hearn, R.A.: Playing games with algorithms: Algorithmic combinatorial game theory. In: Albert, M.H., Nowakowski, R.J. (eds.) Games of No Chance 3. Mathematical Sciences Research Institute Publications, vol. 56, pp. 3–56. Cambridge University Press, Cambridge (2009)Google Scholar
- 7.Eppstein, D.: Computational Complexity of Games and Puzzles (2009), http://www.ics.uci.edu/~eppstein/cgt/hard.html
- 8.Grundy, P.M.: Mathematics and games. Eureka 2, 6–8 (1939)Google Scholar
- 11.Kendall, G., Parkes, A., Spoerer, K.: A Survey of NP-Complete Puzzles. International Computer Games Association Journal 31(1), 13–34 (2008)Google Scholar
- 13.McCarthy, J.: Partial formalizations and the Lemmings game, Technical report, Stanford University, Formal Reasoning Group (1998)Google Scholar
- 15.Robson, J.M.: The complexity of Go. In: Proceedings of the IFIP 9th World Computer Congress on Information Processing, pp. 413–417 (1983)Google Scholar
- 17.Spoerer, K.: The Lemmings Puzzle: Computational Complexity of an Approach and Identification of Difficult Instances. PhD thesis (2007)Google Scholar
- 18.Sprague, R.P.: Ueber mathematische Kampfspiele. Tohoku Mathematical Journal 41, 438–444 (1935/1936)Google Scholar