The Legend of Zelda: The Complexity of Mechanics

TL;DR

This paper investigates the computational complexity of various game mechanics in The Legend of Zelda series, classifying them into complexity classes like polynomial, NP-complete, NP-hard, and PSPACE-complete, and reviews proof techniques used in game complexity analysis.

Contribution

It provides a comprehensive complexity classification of Zelda mechanics and reviews advanced proof techniques for video game complexity over the past decade.

Findings

Certain mechanics are NP-complete or PSPACE-complete.

The paper introduces multiple complexity proof frameworks for video games.

Provides an overview of complexity classifications for Zelda game mechanics.

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.