A 3-approximation algorithm for the facility location problem with uniform capacities

We consider the facility location problem where each facility can serve at most U clients. We analyze a local search algorithm for this problem which uses only the operations of add, delete and swap and prove that any locally optimum solution is no more than 3 times the global optimum. This improves on a result of Chudak and Williamson who proved an approximation ratio of ${3+2\sqrt{2}}$ for the same algorithm. We also provide an example which shows that any local search algorithm which uses only these three operations cannot achieve an approximation guarantee better than 3.