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BusOut is NP-complete

Takehiro Ishibashi, Ryo Yoshinaka, Ayumi Shinohara

TL;DR

This work studies the BusOut puzzle as a computational decision problem by modeling it with a congestion graph $G$, a passenger queue $Q$, and a spot-state $S$. It establishes that deciding solvability, $\ extbf{BusOut}$, is NP-complete even under strong restrictions, via reductions from the $3$-Partition problem, and extends this hardness to general parameter settings using capacity-doubling techniques. It also shows that approximating the minimum number of parking spots is intractable unless $\ ext{P}=\\text{NP}$, while identifying polynomial-time solvable subclasses when congestion graphs are independent and spot counts meet color requirements. The results delineate clear hardness boundaries for puzzle-like scheduling with colored buses and limited parking, informing both theoretical understanding and practical game design.

Abstract

This study examines the computational complexity of the decision problem modeled on the smartphone game Bus Out. The objective of the game is to load all the passengers in a queue onto appropriate buses using a limited number of bus parking spots by selecting and dispatching the buses on a map. We show that the problem is NP-complete, even for highly restricted instances. We also show that it is hard to approximate the minimum number of parking spots needed to solve a given instance.

BusOut is NP-complete

TL;DR

This work studies the BusOut puzzle as a computational decision problem by modeling it with a congestion graph , a passenger queue , and a spot-state . It establishes that deciding solvability, , is NP-complete even under strong restrictions, via reductions from the -Partition problem, and extends this hardness to general parameter settings using capacity-doubling techniques. It also shows that approximating the minimum number of parking spots is intractable unless , while identifying polynomial-time solvable subclasses when congestion graphs are independent and spot counts meet color requirements. The results delineate clear hardness boundaries for puzzle-like scheduling with colored buses and limited parking, informing both theoretical understanding and practical game design.

Abstract

This study examines the computational complexity of the decision problem modeled on the smartphone game Bus Out. The objective of the game is to load all the passengers in a queue onto appropriate buses using a limited number of bus parking spots by selecting and dispatching the buses on a map. We show that the problem is NP-complete, even for highly restricted instances. We also show that it is hard to approximate the minimum number of parking spots needed to solve a given instance.
Paper Structure (4 sections, 8 theorems, 7 equations, 3 figures)

This paper contains 4 sections, 8 theorems, 7 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Formalization of Bus Out
  3. The computational complexity of $\mathbf{BusOut}$
  4. Conclusion and Future Work

Key Result

Theorem 1

Every instance of $\mathbf{BusOut}(\mathcal{B}(s,1,V))$ is solvable for any $s$ and $V$.

Figures (3)

  • Figure 1: Example scenario of Bus Out. (a) initial configuration; (b) deadlock caused by a misstep.
  • Figure 2: Reduction from 3-Partition to $\mathbf{BusOut}(\mathcal{B}(1, 2, \{1\}))$
  • Figure 3: Reduction from $\mathbf{3\textrm{-}Partition}$ to $\mathbf{BusOut}(\mathcal{B}(s, 2, \{1\}))$

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • ...and 6 more