On the computational complexity of a game of cops and robbers

TL;DR

This paper proves that determining the outcome of a perfect-information cops and robbers game on graphs is PSPACE-hard, highlighting its computational complexity and difficulty.

Contribution

It establishes the PSPACE-hardness of deciding the game, resolving a question raised in the 1990s about its computational complexity.

Findings

The game is PSPACE-hard to decide.

The problem remains hard even with simultaneous moves.

It advances understanding of complexity in pursuit-evasion games.

Abstract

We study the computational complexity of a perfect-information two-player game proposed by Aigner and Fromme. The game takes place on an undirected graph where n simultaneously moving cops attempt to capture a single robber, all moving at the same speed. The players are allowed to pick their starting positions at the first move. The question of the computational complexity of deciding this game was raised in the '90s by Goldstein and Reingold. We prove that the game is hard for PSPACE.