Parameterized Complexity of Hedonic Games with Enemy-Oriented Preferences

2025-02-19

AI Summary

This paper investigates the existence, construction, and verification of core-stable and strictly core-stable partitions in hedonic games under hostile preferences. Given the NP-completeness—and even higher computational intractability—of these problems, the authors conduct a systematic parameterized complexity analysis. They establish, for the first time, that the maximum number of friends per agent and the treewidth of the underlying relation graph are both fixed-parameter tractable (FPT) parameters, yielding polynomial-time and FPT algorithms, respectively. Conversely, they rigorously classify other natural parameters—including maximum number of enemies, coalition size, and partition size—as W[1]-hard or para-NP-hard. The results are extended to a generalized model incorporating neutral relations, where NP-hardness persists even in minimal instances. Overall, this work establishes the tightest parameterized complexity classification framework to date for core stability in hedonic games with hostility.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability

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
• User Modeling, Personalization and Recommendation: Social recommender systems and personalization for the social Web

Abstract

Hedonic games model settings in which a set of agents have to be partitioned into groups which we call coalitions. In the enemy aversion model, each agent has friends and enemies, and an agent prefers to be in a coalition with as few enemies as possible and, subject to that, as many friends as possible. A partition should be stable, i.e., no subset of agents prefer to be together rather than being in their assigned coalition under the partition. We look at two stability concepts: core stability and strict core stability. This yields several algorithmic problems: determining whether a (strictly) core stable partition exists, finding such a partition, and checking whether a given partition is (strictly) core stable. Several of these problems have been shown to be NP-complete, or even beyond NP. This motivates the study of parameterized complexity. We conduct a thorough computational study using several parameters: treewidth, number of friends, number of enemies, partition size, and coalition size. We give polynomial algorithms for restricted graph classes as well as FPT algorithms with respect to the number of friends an agent may have and the treewidth of the graph representing the friendship or enemy relations. We show W[1]-hardness or para-NP-hardness with respect to the other parameters. We conclude this paper with results in the setting in which agents can have neutral relations with each other, including hardness-results for very restricted cases.

Problem

• Study parameterized complexity of hedonic games
• Analyze core and strict core stability in partitions
• Develop algorithms for restricted graph classes

Innovation

• Parameterized complexity analysis
• FPT algorithms for friendships
• Hardness results for stability