Relaxed Voronoi: a Simple Framework for Terminal-Clustering Problems
Arnold Filtser, Robert Krauthgamer, Ohad Trabelsi

TL;DR
This paper introduces a simplified framework called Relaxed-Voronoi for solving terminal-clustering problems in metric spaces, providing easier proofs and algorithms for special cases like trees and doubling metrics.
Contribution
The paper shows that the Relaxed-Voronoi framework can be effectively applied to restricted metric spaces, simplifying algorithms and proofs for known bounds.
Findings
Unified framework for multiple terminal-clustering bounds
Simpler algorithms for trees and doubling metrics
Effective use of enlarged, greedy Voronoi cells
Abstract
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 (possibly arising from a graph) and a subset of terminals , and the goal is to partition the points 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Steiner Point Removal on trees [Gupta, SODA 2001], for Metric -Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric -Extension [Englert et al., SICOMP 2014]. A natural approach is to cluster each point with its closest terminal, which would partition into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster…
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