Cost Allocation for Set Covering: the Happy Nucleolus

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, and so that the excess vector is lexicographically maximized. This happy nucleolus has several nice properties. In particular, we show that it can be computed considering a small subset of “simple” coalitions only. While computing the nucleolus for set covering is NP-hard, our results imply that the happy nucleolus can be computed in polynomial time.