A Survey of Algorithms for Capacitated k-median Problems

k-median is a classic optimization problem which has been studied extensively in the last few decades. The current best known approximation ratio for this problem is (2.675 + ). A more generalized version of this problem also considers the capacities of the given facilities. That is, each facility cannot serve more clients than its own capacity. Unfortunately, there is no known constant-factor approximation algorithm for this problem. We survey some state-of-the-art techniques to approximate the Capacitated k-median problem that may violate the hard constraint by a factor of (1 + ). Note that the natural LP relaxation for this problem is still unbounded even when we are allowed to violate the cardinality constraint by a factor of 2− . The new improvements come from several techniques to strengthen the LP as well as a novel clustering algorithm.