Connected k-Center and k-Diameter Clustering

TL;DR

This paper introduces connected k-center and k-diameter clustering problems with connectivity constraints, providing approximation algorithms, optimal solutions for special cases, and complexity bounds, motivated by geodesy applications.

Contribution

It presents the first approximation algorithms for connected clustering problems with connectivity constraints, including improved bounds for Euclidean and doubling metrics, and solutions for special graph cases.

Findings

O(log^2 k)-approximation algorithm for general cases

Constant-factor approximation for Euclidean and doubling metrics

Optimal algorithms for line and tree connectivity graphs

Abstract

Motivated by an application from geodesy, we introduce a novel clustering problem which is a $k$-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph $G$ on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in $G$. We call the resulting problems the connected $k$-center problem and the connected $k$-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an $O(\log^2{k})$-approximation algorithm for the connected $k$-center and the connected $k$-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to $O(1)$. We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a $2$-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.