  We consider a two-stage robust facility location problem on a metric under an
uncertain demand. The decision-maker needs to decide on the (integral) units of
supply for each facility in the first stage to satisfy an uncertain
second-stage demand, such that the sum of first stage supply cost and the
worst-case cost of satisfying the second-stage demand over all scenarios is
minimized. The second-stage decisions are only assignment decisions without the
possibility of adding recourse supply capacity. This makes our model different
from existing work on two-stage robust facility location and set covering
problems. We consider an implicit model of uncertainty with an exponential
number of demand scenarios specified by an upper bound $k$ on the number of
second-stage clients. In an optimal solution, the second-stage assignment
decisions depend on the scenario; surprisingly, we show that restricting to a
fixed (static) fractional assignment for each potential client irrespective of
the scenario gives us an $O(\log k/\log \log k)$-approximation for the problem.
Moreover, the best such static assignment can be computed efficiently giving us
the desired guarantee.
