We consider cost allocation for set covering problems. We allocate as much cost to the elements (players) as possible without violating the group rationality condition (no subset of players pays more than covering this subset would cost), and so that the excess vector is lexicographically maximized. This is identical to the well-known nucleolus if the core of the corresponding cooperative game is nonempty, i.e., if some optimum fractional cover is integral. In general, we call this the 'happy nucleolus'. Like for the nucleolus, the excess vector contains an entry for every subset of players, not only for the sets in the given set covering instance. Moreover, it is NP-hard to compute a single entry because this requires solving a set covering problem. Nevertheless, we give an explicit family of at most $mn$ subsets, each with a trivial cover (by a single set), such that the happy nucleolus is always completely determined by this proxy excess vector; here $m$ and $n$ denote the number of sets and the number of players in our set covering instance. We show that this is the unique minimal such family in a natural sense. While computing the nucleolus for set covering is NP-hard, our results imply that the happy nucleolus can be computed in polynomial time.

翻译：我们考虑集合覆盖问题中的成本分配。我们尽可能多地将成本分配给元素（参与者），同时不违反群体理性条件（任何参与者子集的支付不超过覆盖该子集的成本），并使超额向量按字典序最大化。若对应的合作博弈的核心非空（即存在某个最优分数覆盖为整数覆盖），则此分配与著名的核仁相同。一般情况下，我们称其为“快乐核仁”。与核仁类似，超额向量包含每个参与者子集的对应项，而不仅仅是给定集合覆盖实例中的集合。此外，计算单个项是NP难的，因为这需要求解一个集合覆盖问题。尽管如此，我们给出了一个至多包含$mn$个子集的显式族，每个子集具有平凡覆盖（由单个集合构成），使得快乐核仁始终由该代理超额向量完全决定；这里$m$和$n$分别表示集合覆盖实例中的集合数和参与者数。我们证明这是在自然意义下唯一的最小族。尽管计算集合覆盖的核仁是NP难的，我们的结果表明快乐核仁可在多项式时间内计算。