Generalised Reachability Games

2026-07-15

AI Summary

This study addresses generalized reachability in two-player zero-sum turn-based games on graphs, where Eve aims to visit all sets in a given family of target vertex sets (in any order) while her opponent Adam seeks to prevent this. The authors establish that determining the winner is PSPACE-complete, even when each target set has size at most three, yet becomes fixed-parameter tractable when parameterized by the number of target sets. They provide tight upper and lower bounds on the memory required for winning strategies for both players and demonstrate that most natural optimization variants are undecidable. Decidability can only be recovered under specific commitment models, thereby highlighting the number of target sets as a pivotal parameter governing computational complexity.

Abstract

We study two-player zero-sum turn-based games played on graphs with multiple reachability objectives called generalised reachability games. In classic reachability games the goal of one player, Eve, is to visit a given target set of vertices, and that of the other player, Adam, is to prevent this. In generalised reachability games, the single target set is replaced with a family of target sets and the objective of Eve is to visit all of them in any order. We study the complexity of deciding the winner in two-player games with generalised reachability objectives. Our study reveals that an important parameter that determines the complexity of this problem is the size of the target sets. We first prove that deciding the winner in such games is PSPACE-complete, and the PSPACE lower bound holds even when the size of each target set is at most three. By contrast, we show that the problem is FPT in the number of target sets of size greater than one. Moreover, we consider the memory requirements for both players and give matching upper and lower bounds on the sizes of winning strategies. We also study optimisation variants of these games. For the optimisation problems, we show intractability for most interesting cases. Particularly, in contrast to the tractability of generalised reachability in the case with singleton target sets, the optimisation problem is coNP-hard when Eve tries to maximise the number of target sets that are visited. Tractability of this case can be recovered in a different optimisation setting where Eve is required to pledge a maximum sized subset of target sets that she can guarantee to visit.

Problem

• generalised reachability games
• two-player zero-sum games
• computational complexity
• reachability objectives
• winning strategy

Innovation

• generalised reachability games
• PSPACE-completeness
• fixed-parameter tractability
• memory bounds
• optimisation variants