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.

翻译：分数享乐博弈是一类联盟形成博弈，其中玩家的效用由他们对联盟成员赋予的平均值决定。这类博弈是图享乐博弈的变体，而图享乐博弈则是一类可简洁表示的联盟形成博弈。由于分数享乐博弈在网络聚类中的适用性及其与图享乐博弈的关联，研究者已从多种视角对其进行了广泛探讨。然而，由于效用的非线性特性，在分数享乐博弈中寻找福利最大化的划分是一项具有挑战性的任务。事实上，这一问题已被证明是NP难的，并且仅对有限图类（如树）能在多项式时间内求解。本文提出了（伪）多项式时间算法，用于计算树状图（包括树宽有界图和块图）上分数享乐博弈的福利最大化划分。我们考虑两种社会福利度量：功利主义福利和平等主义福利。此外，文中还给出了一个困难性结果，证明在假设P≠NP的前提下，伪多项式时间可解性已是最优可能。