The Legend of Zelda: The Complexity of Mechanics

Discrete and Computational Geometry, Graphs, and Games

Abstract

We analyze some of the many game mechanics available to Link in the classic Legend of Zelda series of video games. In each case, we prove that the generalized game with that mechanic is polynomial, NP-complete, NP-hard and in PSPACE, or PSPACE-complete. In the process we give an overview of many of the hardness proof techniques developed for video games over the past decade: the motion-planning-through-gadgets framework, the planar doors framework, the doors-and-buttons framework, the "Nintendo" platform game / SAT framework, and the collectible tokens and toll roads / Hamiltonicity framework.