Consider the following problem. There is a set U of n nodes, which we can think of as user...

Consider the following problem. There is a set U of n nodes, which we can think of as users (e.g., these are locations that need to access a service, such as a Web server). You would like to place servers at multiple locations. Suppose you are given a set S possible sites that would be willing to act as locations for the servers. For each site s e S, there is a fee fs > 0 for placing a server at that location. Your goal will be to approximately minimize the cost while providing the service to each of the customers. So far this is very much like the Set Cover Problem: The places s are sets, the weight of set s is fs, and we want to select a collection of sets that covers all users. There is one extra complication: Users u ε U can be served from multiple sites, but there is an associated cost dus for serving user u from site s. When the value dus is very high, we do not want to serve user u from site s; and in general the service cost dus serves as an incentive to serve customers from "nearby" servers whenever possible.

So here is the question, which we call the Facility Location Problem: Given the sets U and S, and costs f and d, you need to select a subset A c S at which to place servers (at a cost of ∑sεAfs), and assign each user u to the active server where it is cheapest to be served, minsεA dus. The goal is to minimize the overall cost . Give an H(n)-approximation for this problem.

(Note that if all service costs dus are 0 or infinity, then this problem is exactly the Set Cover Problem: fs is the cost of the set named s, and dus is 0 if node u is in set s, and infinity otherwise.)