The Complexity of Graph Exploration Games

TL;DR

This paper investigates the computational complexity of graph exploration and treasure hunt problems with an unlabeled map, revealing PSPACE-completeness across various settings and constraints, highlighting their inherent difficulty.

Contribution

It introduces decision variants of exploration and treasure hunt problems with an unlabeled map and proves their PSPACE-completeness in multiple scenarios.

Findings

Most of these exploration and treasure hunt games are PSPACE-complete.

Complexity persists across directed and undirected graphs with various constraints.

Additional solution constraints do not significantly reduce the computational difficulty.

Abstract

Graph Exploration problems ask a searcher to explore an unknown environment. The environment is modeled as a graph, where the searcher needs to visit each vertex beginning at some vertex. Treasure Hunt problems are a variation of Graph Exploration, in which the searcher needs to find a hidden treasure, which is located at a designated vertex. Usually these problems are modeled as online problems, and any online algorithm performs poorly because it has too little knowledge about the instance to react adequately to the requests of the adversary. Thus, the impact of a priori knowledge is of interest. One form of a priori knowledge is an unlabeled map, which is an isomorphic copy of the graph. We analyze Graph Exploration and Treasure Hunt problems with an unlabeled map that is provided to the searcher. For this, we formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is whether the searcher is able to explore the graph completely or find the treasure for all possible decisions of the adversary. We analyze these games in multiple settings, with and without costs on the edges, on directed and undirected graphs and with different constraints (allowing multiple visits to vertices or edges) on the solution. We prove PSPACE-completeness for most of these games. Additionally, we analyze the complexity of related problems that have additional constraints on the solution.