Maximizing Utilitarian and Egalitarian Welfare of Fractional Hedonic Games on Tree-like Graphs

Fractional hedonic games are coalition formation games where a player's utility is determined by the average value they assign to the members of their coalition. These games are a variation of graph hedonic games, which are a class of coalition formation games that can be succinctly represented. Due to their applicability in network clustering and their relationship to graph hedonic games, fractional hedonic games have been extensively studied from various perspectives. However, finding welfare-maximizing partitions in fractional hedonic games is a challenging task due to the nonlinearity of utilities. In fact, it has been proven to be NP-hard and can be solved in polynomial time only for a limited number of graph classes, such as trees. This paper presents (pseudo)polynomial-time algorithms to compute welfare-maximizing partitions in fractional hedonic games on tree-like graphs. We consider two types of social welfare measures: utilitarian and egalitarian. Tree-like graphs refer to graphs with bounded treewidth and block graphs. A hardness result is provided, demonstrating that the pseudopolynomial-time solvability is the best possible under the assumption P$\neq$NP.