We study the recently introduced fair division concept of the happy nucleolus for cost allocation among players in a cooperative game, with special focus on its computation. The happy nucleolus applies the same fairness criterion as the well-established nucleolus but with reduced total value. Still, we show that the relation between the two concepts is quite involved, and intuitive properties do not hold - e.g., the entry of a player in the happy nucleolus can be larger than the entry of the same player in the nucleolus, even for monotone and subadditive games. This refutes conjectures of Meir, Rosenschein and Malizia (2011). Further, we study the separation problem of the linear programs appearing in the MPS scheme for computing the (happy) nucleolus. It includes linear subspace avoidance constraints, which can be handled efficiently for problems with a certain dynamic programming formulation due to Köhnemann and Toth (2020). We show how to get rid of these constraints for all monotone games if we allow for an arbitrarily small error of epsilon, thus conserving known approximation guarantees for the same problem without subspace avoidance. Finally, we focus on practical results at the example of vehicle routing games by designing an efficient heuristic based on our previous insights and past work, and demonstrate its power.

翻译：我们研究了最近提出的合作博弈中成本分摊的公平分配概念——快乐核仁，特别关注其计算方法。快乐核仁采用与已确立的核仁相同的公平性准则，但总价值有所降低。尽管如此，我们证明这两个概念之间的关系相当复杂，且直观性质并不成立——例如，即使对于单调且次可加博弈，玩家在快乐核仁中的分配值也可能大于其在核仁中的分配值。这反驳了Meir、Rosenschein和Malizia（2011）的猜想。进一步，我们研究了用于计算（快乐）核仁的MPS方案中出现的线性规划分离问题。该问题包含线性子空间规避约束，得益于Köhnemann和Toth（2020）的研究，对于具有特定动态规划形式的问题，这些约束可被高效处理。我们证明，若允许任意小的误差ε，则对于所有单调博弈，可以消除这些约束，从而保留无子空间规避时相同问题的已知近似保证。最后，我们以车辆路径博弈为例，基于先前见解和已有工作设计了一种高效启发式算法，并展示了其有效性。