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We introduce a problem that is a common generalization of the uncapacitatedfacility location and minimum latency (ML) problems, where facilities need tobe opened to serve clients and also need to be sequentially activated beforethey can provide service. Formally, we are given a set \F of n facilities withfacility-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 rdenoting the depot, and a time metric d on \F\cup{r}. Our goal is to open asubset F of facilities, find a path P starting at r and spanning F to activatethe open facilities, and connect each client j to a facility \phi(j)\in F, soas to minimize \sum_{i\in F}f_i +\sum_{clients j}(c_{\phi(j),j}+t_j), where t_jis the time taken to reach \phi(j) along path P. We call this the minimumlatency 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 animprovement 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 thetime metric); (b) metric uniform MLUFL (metric connection costs, unformtime-metric). Our LP-based methods are versatile and easily adapted to yieldapproximation 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 thefacility to which it is connected; and (ii) the k-route version, where kvehicles are routed in parallel to activate the open facilities. Our LP-basedunderstanding of MLUFL also offers some LP-based insights into ML, which webelieve is a promising direction for obtaining improvements for ML.

Facility Location with Client Latencies: Linear-Programming based Techniques for Minimum-Latency Problems