We consider the classical vertex cover and set cover problems with the addition of hard capacity constraints. This means that a set (vertex) can only cover a limited number of its elements (adjacent edges) and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems that also captures resource limitations in practical scenarios.

We obtain the following results. For the unweighted vertex cover problem with hard capacities we give a 3-approximation algorithm which is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem. This is an interesting separation between the approximability of weighted and unweighted versions of a "natural" graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey [23] on submodular set cover. We provide in this paper a simple and intuitive proof for this bound.