A cooperative game is defined as a set of players and a cost function. The distribution of the whole cost between theplayers can be done using the core concept, that is the set of all undominated cost allocations which prevent playersfrom grouping. In this paper we study a game whose cost function comes from the optimal solution of a linear integercovering problem. We give necessary and sufficient conditions for the core to be nonempty and characterize itsallocations using linear programming duality. We also discuss a special allocation, called the nucleolus. Wecharacterize that allocation and show that it can be computed in polynomial time using a column generation method.