Abstract: 
We consider the problem of finding the optimal coalition structure in \emph{Weighted Graph Games (WGGs)}, a simple restricted representation of coalitional games [Deng, Papadimitriou, Math. OR 1994]. The agents in such games are vertices of the graph, and the value of a coalition is the sum of the weights of the edges present between coalition members. The \emph{optimal coalition structure} is a partition of the agents to coalitions, that maximizes the sum of utilities obtained by the coalitions. We show that finding the optimal coalition structure is not only hard for general graphs, but is also intractable for restricted families such as planar graphs which are amenable for many other combinatorial problems. We then provide algorithms with constant factor approximations for planar, minorfree and bounded degree graphs.
