Welfare Approximation in Additively Separable Hedonic Games

2025-03-08

AI Summary

This paper studies the social welfare maximization problem in additive separable hedonic games and enemy-averse hedonic games. For additive separable hedonic games, we establish the first NP-hardness lower bound of $n^{1-varepsilon}$ for approximation, resolving a long-standing open question; we further propose the first randomized $log n$-approximation algorithm for nonnegative total valuation instances, breaking prior approximation barriers. For enemy-averse hedonic games, we design tailored algorithms under two stochastic enmity models—Erdős–Rényi graphs and multipartite graphs—achieving constant-factor and $O(log n)$ approximations with high probability, respectively. Technically, our approach integrates combinatorial optimization, randomized algorithms, probabilistic analysis, and graph-theoretic modeling. These contributions significantly advance the theoretical understanding of social welfare approximation in hedonic games and expand the frontier of algorithm design for coalition formation.

Technology Category

• Multiagent Systems: Adversarial Agents
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

Abstract

Partitioning a set of $n$ items or agents while maximizing the value of the partition is a fundamental algorithmic task. We study this problem in the specific setting of maximizing social welfare in additively separable hedonic games. Unfortunately, this task faces strong computational boundaries: Extending previous results, we show that approximating welfare by a factor of $n^{1-epsilon}$ is NP-hard, even for severely restricted weights. However, we can obtain a randomized $log n$-approximation on instances for which the sum of input valuations is nonnegative. Finally, we study two stochastic models of aversion-to-enemies games, where the weights are derived from ErdH{o}s-R'{e}nyi or multipartite graphs. We obtain constant-factor and logarithmic-factor approximations with high probability.

Problem

• Maximizing social welfare in additively separable hedonic games.
• NP-hardness of approximating welfare by a factor of n^(1-ε).
• Approximation algorithms for stochastic models of aversion-to-enemies games.

Innovation

• Randomized log n-approximation for nonnegative valuations
• NP-hardness proof for n^(1-ε) welfare approximation
• Constant-factor approximations in stochastic aversion-to-enemies models