  In the k-center problem, given a metric space V and a positive integer k, one
wants to select k elements (centers) of V and an assignment from V to centers,
minimizing the maximum distance between an element of V and its assigned
center. One of the most general variants is the capacitated
{\alpha}-fault-tolerant k-center, where centers have a limit on the number of
assigned elements, and, if {\alpha} centers fail, there is a reassignment from
V to non-faulty centers. In this paper, we present a new approach to tackle
fault tolerance, by selecting and pre-opening a set of backup centers, then
solving the obtained residual instance. For the {0,L}-capacitated case, we give
approximations with factor 6 for the basic problem, and 7 for the so called
conservative variant, when only clients whose centers failed may be reassigned.
Our algorithms improve on the previously best known factors of 9 and 17,
respectively. Moreover, we consider the case with general capacities. Assuming
{\alpha} is constant, our method leads to the first approximations for this
case. We also derive approximations for the capacitated fault- tolerant
k-supplier problem.
