  We introduce a problem that is a common generalization of the uncapacitated
facility location and minimum latency (ML) problems, where facilities need to
be opened to serve clients and also need to be sequentially activated before
they can provide service. Formally, we are given a set \F of n facilities with
facility-opening costs {f_i}, a set of m clients, and connection costs {c_{ij}}
specifying the cost of assigning a client j to a facility i, a root node r
denoting the depot, and a time metric d on \F\cup{r}. Our goal is to open a
subset F of facilities, find a path P starting at r and spanning F to activate
the open facilities, and connect each client j to a facility \phi(j)\in F, so
as to minimize \sum_{i\in F}f_i +\sum_{clients j}(c_{\phi(j),j}+t_j), where t_j
is the time taken to reach \phi(j) along path P. We call this the minimum
latency uncapacitated facility location (MLUFL) problem.
  Our main result is an O(\log n\max{\log n,\log m})-approximation for MLUFL.
We also show that any improvement in this approximation guarantee, implies an
improvement in the (current-best) approximation factor for group Steiner tree.
We obtain constant approximations for two natural special cases of the problem:
(a) related MLUFL (metric connection costs that are a scalar multiple of the
time metric); (b) metric uniform MLUFL (metric connection costs, unform
time-metric). Our LP-based methods are versatile and easily adapted to yield
approximation guarantees for MLUFL in various more general settings, such as
(i) when the latency-cost of a client is a function of the delay faced by the
facility to which it is connected; and (ii) the k-route version, where k
vehicles are routed in parallel to activate the open facilities. Our LP-based
understanding of MLUFL also offers some LP-based insights into ML, which we
believe is a promising direction for obtaining improvements for ML.
