The Power of Matching for Online Fractional Hedonic Games

We study coalition formation in the framework of fractional hedonic games (FHGs). The objective is to maximize social welfare in an online model where agents arrive one by one and must be assigned to coalitions immediately and irrevocably. For general online FHGs, it is known that computing maximal matchings achieves the optimal competitive ratio, which is, however, unbounded for unbounded agent valuations. We achieve a constant competitive ratio in two related settings while carving out further connections to matchings. If algorithms can dissolve coalitions, then the optimal competitive ratio of $\frac{1}{6+4\sqrt{2}}$ is achieved by a matching-based algorithm. Moreover, we perform a tight analysis for the online matching setting under random arrival with an unknown number of agents. This entails a randomized $\frac 16$-competitive algorithm for FHGs, while no algorithm can be better than $\frac 13$-competitive.