Research Proposal Fully Dynamic Approximate Clustering

Given a metric (U, d) with |U | = n, a subset S ⊆ U with |S| = l, a non-negative function w : U → <, and an integer k ≤ l, the generalized static k-center and k-median problems ask to pick a subset X ⊆ S with |X| = k so as to minimize maxx∈U d(x,X) and ∑ x∈U d(x,X) · w(x), respectively, where d(x,X) = miny∈X d(x, y). Each point in X is called a center and each point in U is assigned to its closest center. Each center together with all nodes assigned to it form a cluster. Classic versions of these problems assume S = U . Both problems are NP-hard, but efficient constant-factor approximation algorithms exist [1, 8, 10]. The fully dynamic versions of the problems require us to maintain a constant-factor approximate solution to the corresponding problem under dynamic updates (insertion and deletion of points) to the set U .