Hardness of Facility Location Problems

1.2 Hardness of k-median Theorem 1. It is hard to approximate k-median within 1+ 2 e1/c for any c < 1. This theorem is equivalent to the 1− 1/e− hardness shown in [2] and follows from a standard reduction from set cover. The main idea is to take an approximation algorithm for k-median, and use it to obtain a partial set cover. Then repeat this process, again partially covering the remaining items, until everything is covered. A k-median approximation ratio of 1 + 2e is just enough to obtain a set cover approximation of lnn, which is the best possible under reasonable hardness assumptions. Any better approximation to k-median violates this.