Relaxed Voronoi: A Simple Framework for Terminal-Clustering Problems

We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space $(X,d)$ (possibly arising from a graph) and a subset of terminals $K\subset X$, and the goal is to partition the points $X$ such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric $0$-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric $0$-Extension [Englert et al., SICOMP 2014]. A natural approach is to cluster each point with its closest terminal, which would partition $X$ into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.