Welfare Approximation in Additively Separable Hedonic Games
Martin Bullinger, Vaggos Chatziafratis, Parnian Shahkar
TL;DR
The paper studies welfare maximization in additively separable hedonic games (ASHGs), proving strong worst-case hardness and identifying tractable regimes. It shows an $n^{1-\varepsilon}$-inapproximability for restricted valuations and, when the total value is nonnegative, a randomized $O(\log n)$-approximation by connecting welfare to correlation clustering. It then analyzes stochastic models based on Erdős–Rényi and random balanced/multipartite graphs, obtaining constant-factor and $O(\log n)$-approximation guarantees with high probability via greedy methods and Turán-graph reductions. Overall, the work clarifies the boundary between hard and tractable instances of welfare optimization in hedonic games and highlights algorithmic techniques that generalize to related graph-partitioning problems.
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 $n^{1-ε}$ 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 Erdős-Rényi or multipartite graphs. We obtain constant-factor and logarithmic-factor approximations with high probability.