Temporal Explorability Games

TL;DR

This paper studies the complexity of exploration games on temporal graphs, revealing that temporal dynamics significantly increase computational difficulty compared to static graphs.

Contribution

It characterizes the computational complexity of explorability games on temporal graphs, showing NP-completeness and PSPACE-completeness results, and extends these results to symbolic representations.

Findings

Explorability on static graphs is as hard as reachability.

On temporal graphs, explorability matches generalized reachability complexity.

Symbolic temporal graphs make the problem PSPACE-hard.

Abstract

Temporal graphs extend ordinary graphs with discrete time that affects the availability of edges. We consider solving games played on temporal graphs where one player aims to explore the graph, i.e., visit all vertices. The complexity depends majorly on two factors: the presence of an adversary and how edge availability is specified. We demonstrate that on static graphs, where edges are always available, solving explorability games is just as hard as solving reachability games. In contrast, on temporal graphs, the complexity of explorability coincides with generalized reachability (NP-complete for one-player and PSPACE- complete for two player games). We further show that if temporal graphs are given symbolically, even one-player reachability and thus explorability and generalized reachability games are PSPACE-hard. For one player, all these are also solvable in PSPACE and for two players, they are in PSPACE, EXP and EXP, respectively.