Computational Complexity of Game Boy Games

TL;DR

This work analyzes the generalized computational complexity of several Game Boy games, proving $NP$-hardness for generalized Donkey Kong, Wario Land, Harvest Moon GB, Mole Mania, and related titles through polynomial-time Karp reductions from classic $NP$-complete problems such as $3$-CNF-$Sat$, $Sat$, $Hamiltonian Cycle$, and $Knapsack$. It introduces formal definitions (Game Room, Game Transition, Game Level) and employs game-specific gadgets—switches, slide boards, skull doors, one-way transitions, and Push-1 mappings—to encode computational constraints. The authors also discuss known NP-hard games and easily derived reductions, and highlight open problems, including PSPACE-hardness and the complexity of Dr. Mario. Overall, the paper establishes the NP-hardness of several generalized Game Boy games and contributes a detailed NP-hardness proof for Harvest Moon GB, underscoring the computational intractability of certain game-level design problems under broad assumptions.

Abstract

We analyze the computational complexity of several popular video games released for the Nintendo Game Boy video game console. We analyze the complexity of generalized versions of four popular Game Boy games: Donkey Kong, Wario Land, Harvest Moon GB, and Mole Mania. We provide original proofs showing that these games are \textbf{NP}-hard. Our proofs rely on Karp reductions from four of Karp's original 21 \textbf{NP}-complete problems: \textsc{Sat}, \textsc{3-Cnf-Sat}, \textsc{Hamiltonian Cycle}, and \textsc{Knapsack}. We also discuss proofs easily derived from known results demonstrating the \textbf{NP}-hardness of Lock `n' Chase and The Lion King.

Figures (5)

• Figure 1: Any 3-CNF-Sat instance can be converted to a level in Donkey Kong in polynomial time
• Figure 2: Wario Land contains a door and key game mechanic.
• Figure 3: One-way paths can be constructed in Wario Land
• Figure 4: Crop planting and harvesting mechanics of Harvest Moon GB
• Figure 5: The game mechanics of Mole Mania. Left: the first-floor tile from the top-left (after the water) is a Hard tile. The third-floor tile from the top-left is a Soft tile. Right: Weight tiles are special tiles in Mole Mania that can only be pushed, but not pulled.

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 2.1
• Theorem 2.2
• proof
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6