Maximizing the Egalitarian Welfare in Friends and Enemies Games

We consider the complexity of maximizing egalitarian welfare in Friends and Enemies Games -- a subclass of hedonic games in which every agent partitions other agents into friends and enemies. We investigate two classic scenarios proposed in the literature, namely, Friends Appreciation ($\mathsf{FA}$) and Enemies Aversion ($\mathsf{EA}$): in the former, each agent primarily cares about the number of friends in her coalition, breaking ties based on the number of enemies, while in the latter, the opposite is true. For $\mathsf{EA}$, we show that our objective is hard to approximate within $O(n^{1-\epsilon})$, for any fixed $\epsilon>0$, and provide a polynomial-time $(n-1)$-approximation. For $\mathsf{FA}$, we obtain an NP-hardness result and a polynomial-time approximation algorithm. Our algorithm achieves a ratio of $2-\Theta(\frac{1}{n})$ when every agent has at least two friends; however, if some agent has at most one friend, its approximation ratio deteriorates to $n/2$. We recover the $2-\Theta(\frac{1}{n})$ approximation ratio for two important variants: when randomization is allowed and when the friendship relationship is symmetric. Additionally, for both $\mathsf{EA}$ and $\mathsf{FA}$ we identify special cases where the optimal egalitarian partition can be computed in polynomial time.