Facility Location with Client Latencies

We introduce a problem that is a common generalization of the uncapacitated facility location (UFL) and minimum latency (ML) problems, where facilities not only need to be opened to serve clients, but also need to be sequentially activated before they can provide service. This abstracts a setting where inventory demanded by customers needs to be stocked or replenished at facilities from a depot or warehouse. Formally, we are given a set ℱ of n facilities with facility-opening costs, a set 𝒟 of m clients, and connection costs c ( i , j ) 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 ∪ { r }. Our goal is to open a subset of facilities, find a path P starting at r and spanning the open facilities to activate them, and connecting each client j to an open facility so as to minimize the total facility opening cost, the total client connection cost, and the total time of arrivals at each facility along 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. Via a reduction to the group Steiner tree (GST) problem, we show this result is tight in the sense that any improvement in the approximation guarantee for MLUFL, implies an improvement in the (currently known) approximation factor for GST. We obtain significantly improved constant approximation guarantees for two natural special cases of the problem: (a) Related MLUFL, where the connection costs form a metric that is a scalar multiple of the time metric; (b) Metric uniform MLUFL, where we have metric connection costs and the time-metric is uniform. Our LP-based methods are fairly versatile and are easily adapted with minor changes to yield approximation guarantees for MLUFL (and ML) in various more general settings, such as (i) the setting where the latency-cost of a client is a function (of bounded growth) of the delay faced by the facility to which it is connected; and (ii) the k -route version, where we can dispatch k vehicles in parallel to activate the open facilities. Our LP-based understanding of MLUFL also offers some LP-based insights into ML. We obtain two natural LP-relaxations for ML with constant integrality gap, which we believe shed new light upon the problem and offer a promising direction for obtaining improvements for ML.