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 (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.