Incremental Algorithms for Facility Location and k-Median

In the incremental versions of Facility Location and k-Median, the demand points arrive one at a time and the algorithm must maintain a good solution by either adding each new demand to an existing cluster or placing it in a new singleton cluster. The algorithm can also merge some of the existing clusters at any point in time. We present the first incremental algorithm for Facility Location which achieves a constant performance ratio and the first incremental algorithm for k-Median which achieves a constant performance ratio using O(k) medians, thus resolving an open question of [7]. The algorithm is based on a novel merge rule which ensures that the algorithm’s configuration monotonically converges to the optimal facility locations according to a certain notion of distance. Using this property, we reduce the general case to the special case that the optimal solution consists of a single facility.