BusOut is NP-complete

2025-05-14

AI Summary

This paper investigates the bus scheduling decision problem modeled by the smartphone game *Bus Out*, which involves assigning passenger queues to compatible buses and dispatching them under limited parking capacity. The authors first establish that the problem remains NP-complete even under highly restricted conditions—namely, on a path-shaped network, with unit passenger capacity per bus, and fixed departure times. Subsequently, via a polynomial-time reduction from 3-SAT, they prove that minimizing the required number of parking spaces is APX-hard, implying no polynomial-time algorithm can achieve an arbitrary constant-factor approximation. This work constitutes the first computational complexity analysis of a transportation-themed casual game, bridging a gap in the theoretical understanding of such games. It provides foundational insights for both game difficulty design and the theoretical limits of algorithmic solutions to real-world transit scheduling variants.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Search

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Determining BusOut game's NP-complete complexity
• Loading passengers onto buses with limited parking
• Hard to approximate minimum required parking spots

Innovation

• Proves BusOut problem is NP-complete
• Analyzes restricted instances complexity
• Hard to approximate parking spots