Covering Problems with Hard Capacities

We consider the classical vertex cover and set cover problems with hard capacity constraints. This means that a set (vertex) can cover only 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 which 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 that is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem, yielding 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 [Combinatorica, 2 (1982), pp. 385-393] on submodular set cover. We provide here a simple and intuitive proof for this bound.