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Games on Graphs: A Time-Efficient Algorithm for Solving Finite Reachability and Safety Games

Christian Giannetti

TL;DR

This paper tackles the problem of solving finite reachability and safety games on graphs by exploiting the duality between these game types to design a time-efficient, multiple-perspective algorithm. The approach blends forward and backward solving strategies, leverages transpose graphs, and incorporates several optimizations (Current set, Processed list, and hybrid graph checks) to accelerate attractor-based computations and enforce uniform memoryless winning strategies in $O(|V|+|E|)$ time for the winning regions. Experimental evaluation on randomized graphs demonstrates that the multiple-perspective algorithm often outperforms naive forward and backward methods, with substantial speedups in medium-to-large graphs, albeit with overheads from transpose graph construction and viewpoint switching in some cases. The work provides a scalable, practical method for games on graphs that can impact applications in economics, politics, and epidemiology, and sets the stage for extending the framework to more complex game dynamics.

Abstract

In recent years, there has been a growing interest in games on graphs within the research community, fueled by their relevance in applications such as economics, politics, and epidemiology. This paper aims to comprehensively detail the design decisions involved in developing a time-efficient algorithm for solving finite reachability and safety games on graphs. The primary contribution of this work is the introduction of a novel algorithm that effectively addresses both reachability and safety games by exploiting their inherent duality. The performance of the proposed algorithm is rigorously evaluated against traditional methods using a randomized testing framework. The paper is organized as follows: first, we provide the reader with a theoretical overview of reachability and safety games, followed by an in-depth discussion on the construction of the playing arena. A formal definition of reachability and safety games and a review of traditional algorithms for their resolution are then presented. Subsequently, the multiple-perspective algorithm is introduced along with its optimizations. The paper concludes with an extensive set of experiments and a comprehensive discussion of their results.

Games on Graphs: A Time-Efficient Algorithm for Solving Finite Reachability and Safety Games

TL;DR

This paper tackles the problem of solving finite reachability and safety games on graphs by exploiting the duality between these game types to design a time-efficient, multiple-perspective algorithm. The approach blends forward and backward solving strategies, leverages transpose graphs, and incorporates several optimizations (Current set, Processed list, and hybrid graph checks) to accelerate attractor-based computations and enforce uniform memoryless winning strategies in time for the winning regions. Experimental evaluation on randomized graphs demonstrates that the multiple-perspective algorithm often outperforms naive forward and backward methods, with substantial speedups in medium-to-large graphs, albeit with overheads from transpose graph construction and viewpoint switching in some cases. The work provides a scalable, practical method for games on graphs that can impact applications in economics, politics, and epidemiology, and sets the stage for extending the framework to more complex game dynamics.

Abstract

In recent years, there has been a growing interest in games on graphs within the research community, fueled by their relevance in applications such as economics, politics, and epidemiology. This paper aims to comprehensively detail the design decisions involved in developing a time-efficient algorithm for solving finite reachability and safety games on graphs. The primary contribution of this work is the introduction of a novel algorithm that effectively addresses both reachability and safety games by exploiting their inherent duality. The performance of the proposed algorithm is rigorously evaluated against traditional methods using a randomized testing framework. The paper is organized as follows: first, we provide the reader with a theoretical overview of reachability and safety games, followed by an in-depth discussion on the construction of the playing arena. A formal definition of reachability and safety games and a review of traditional algorithms for their resolution are then presented. Subsequently, the multiple-perspective algorithm is introduced along with its optimizations. The paper concludes with an extensive set of experiments and a comprehensive discussion of their results.
Paper Structure (23 sections, 13 equations, 2 figures, 4 tables, 6 algorithms)

This paper contains 23 sections, 13 equations, 2 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Deterministic turn-based games
  3. Reachability and safety games
  4. Representing the graph: data structures employed
  5. Random graph generation: automatic experiments
  6. Random graph generation: avoiding isolated nodes
  7. From theory to practice: problem formulation from different perspectives
  8. Safety-reachability duality: Reachability problem formulation
  9. Safety-reachability duality: naive purely forward algorithm
  10. Optimizations
  11. Transpose graph optimization
  12. Transpose graph optimization: simultaneous usage of multiple graphs
  13. Transpose graph optimization: applicative example
  14. Optimizations for reachability games
  15. Optimizations for reachability games: Current set optimization
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Graphical representation of the given arena. The round-shaped nodes are the nodes controlled by the reachability player, while the square-shaped nodes indicate the nodes controlled by the safety player. Lastly, the region colored in green identifies the reachability of the player's target set.
  • Figure 2: Graphical representation of the functioning of the Current set optimization.