Closing the competitive-ratio gap for FHGs under random arrival

Determine the exact optimal competitive ratio for online social welfare maximization in fractional hedonic games when agents arrive in a uniformly random order and coalition assignments are irrevocable (i.e., without free dissolution), by closing the gap between the existing 1/6-competitive algorithm derived from online matching and the 1/3 upper bound established for this model.

Background

The paper studies online coalition formation in fractional hedonic games (FHGs) and establishes strong connections to online maximum weight matching (MWM). Under random arrival with an unknown number of agents, the authors provide a randomized algorithm achieving a 1/6-competitive ratio for FHGs by leveraging a matching-based approach and the known result that every maximum weight matching is a 1/2-approximation of social welfare in symmetric FHGs.

They also prove that no algorithm can be better than 1/3-competitive under random arrival by establishing a tight lower bound in the matching domain on the tree domain and transferring this limitation to FHGs. Consequently, there is a gap between the 1/6 achievable algorithm and the 1/3 impossibility bound in the FHG setting under random arrival without free dissolution, which the authors explicitly leave as an open problem to resolve.