  We study a variant of Set Cover where each element of the universe has some
demand that determines how many times the element needs to be covered.
Moreover, we examine two generalizations of this problem when a set can be
included multiple times and when sets have different prices. We prove that all
three problems are fixed-parameter tractable with respect to the combined
parameter budget, the maximum number of elements missing in one of the sets,
and the maximum number of sets in which one of the elements does not occur.
Lastly, we point out how our fixed-parameter tractability algorithm can be
applied in the context of bribery for the (collective-decision) group
identification problem.
