Computational Complexity of Ensuring Individual Rationality in Topological Distance Games

Core Concepts

Ensuring individual rationality, a minimal stability requirement, is computationally intractable in topological distance games even in very basic cases.

Abstract

The paper presents a comprehensive study of the computational complexity of ensuring individual rationality (IR) in topological distance games. Topological distance games model scenarios where the utility of an agent depends on both its inherent preferences for other agents and its distance from them on an underlying topology.
The key findings are:

Ensuring IR is NP-complete even with very restricted utilities and topologies. This holds true regardless of the distance factor function used.

The problem remains intractable (W[1]-hard) when parameterized by the number of agents, and this is tight as an XP algorithm exists.

Restricting the topology alone is not sufficient for tractability - the problem remains W[1]-hard even on path topologies.

Combining restrictions on the enmity graph (at most one agent has enemies) and the topology (path) yields fixed-parameter tractability.

Further positive results show fixed-parameter tractability when parameterized by the number of agents and either the twin-width or shrub-depth of the topology.

The paper establishes the precise boundaries between tractable and intractable cases for the problem of ensuring individual rationality in topological distance games.

Stats

The sum of all elements in the Equitable Partition problem instance is 2k.
The minimum value of any element in the Equitable Partition problem instance is at least n^2.
The maximum difference between any two elements in the Equitable Partition problem instance is at most min(S) / n^2.

Quotes

"Ensuring individual rationality, a minimal requirement for a solution to be considered stable, is computationally intractable in topological distance games even in very basic cases."
"To reach at least some tractability, one needs to combine multiple restrictions of the input instance, including the number of agents and the topology and the influence of distant agents on the utility."

Key Insights Distilled From

Individual Rationality in Topological Distance Games is Surprisingly Hard

by Argy... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.14128.pdf

Deeper Inquiries

How can the techniques developed in this paper be extended to study other solution concepts beyond individual rationality in topological distance games

The techniques developed in this paper for studying individual rationality in topological distance games can be extended to explore other solution concepts, such as stability and fairness. One possible extension is to investigate the concept of stability in these games, where agents have preferences over their assignments and may have incentives to deviate if they can improve their utility. By analyzing different stability notions like core stability or Nash stability in the context of topological distance games, researchers can gain insights into the robustness and equilibrium properties of these games. Additionally, exploring fairness criteria, such as envy-freeness or Pareto efficiency, can provide a deeper understanding of the social welfare implications of the assignments in topological distance games. By incorporating these solution concepts into the analysis, researchers can offer a more comprehensive evaluation of the outcomes and properties of these games.

What are the implications of the hardness results on the practical applicability of topological distance games in real-world scenarios

The hardness results presented in this paper have significant implications for the practical applicability of topological distance games in real-world scenarios. The complexity of finding individually rational outcomes, as demonstrated to be intractable even in basic cases, suggests that computing stable and desirable assignments in these games can be highly challenging. This complexity may limit the scalability and efficiency of using topological distance games in practical settings where quick and reliable decision-making is crucial. The intractability results also highlight the need for developing efficient algorithms and heuristics to handle the computational challenges posed by these games. Despite these challenges, understanding the computational complexity of topological distance games can guide the design of better algorithms and strategies for addressing similar problems in real-world applications.

Can the fixed-parameter tractable algorithms be further improved in terms of their running time complexity

While the fixed-parameter tractable algorithms presented in the paper provide a promising approach to handle the computational complexity of topological distance games, there is room for further improvement in terms of their running time complexity. One potential avenue for enhancing the efficiency of these algorithms is to explore advanced algorithmic techniques, such as dynamic programming, branch and bound, or kernelization, to optimize the parameterized algorithms. By refining the algorithmic design and incorporating more sophisticated data structures and optimization strategies, researchers can potentially reduce the running time of the fixed-parameter tractable algorithms for topological distance games. Additionally, leveraging parallel computing or distributed computing frameworks can help expedite the computation process and enhance the scalability of these algorithms for larger instances of topological distance games. By continuously refining and optimizing the algorithms, researchers can strive to achieve faster and more efficient solutions for topological distance games.