Temporal Explorability Games

2024-12-20

arXiv.org

AI Summary

This paper investigates the computational complexity of exploratory games on temporal graphs, where edges become available dynamically over time and one or two players must visit all vertices. The presence or absence of an adversary and the input representation of edge availability—explicit (listing all time steps) versus symbolic (using compact logical formulas)—critically affect problem hardness. We establish the first systematic complexity classification for such games, proving their equivalence to generalized temporal reachability. Under explicit input encoding, the single-player variant is NP-complete and the two-player variant PSPACE-complete; under symbolic encoding, both become PSPACE-hard. Crucially, all variants are strictly contained in PSPACE. Our main contribution is a unified complexity framework for temporal graph exploration games, which precisely delineates how input representation determines tractability boundaries. This work provides foundational insights for formal verification of temporal systems, linking game-theoretic exploration to core problems in temporal logic and automata theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Temporal Planning
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Explorability games on temporal graphs with discrete time.
• Complexity depends on adversary presence and edge availability.
• One-player games are PSPACE-hard and solvable in PSPACE.

Innovation

• Temporal graphs extend static graphs with time.
• Explorability games complexity depends on adversary presence.
• Symbolic temporal graphs make reachability games PSPACE-hard.