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$k$-Center Clustering in Distributed Models

Leyla Biabani, Ami Paz

TL;DR

This work initiates the study of k-center in distributed models where the network graph defines the metric via shortest paths, examining LOCAL, CONGEST, and CLIQUE settings. It develops a spectrum of results: a simple (2+ε)k-approximation in the LOCAL model and a tight linear-time barrier for better-than-(k−1) performance, a 2-approximation in CONGEST using BFS-based greedy techniques, and a near-all-pairs-shortest-path–driven 2-approximation in the CLIQUE model with varying trade-offs dependent on distance computation accuracy. The paper also establishes strong lower bounds in CONGEST via reductions from disjointness, extended to k-center, and discusses the challenges of lower bounds in the CLIQUE model. Collectively, the results delineate the distributed complexity landscape of graph-metric k-center, highlighting the role of communication constraints and metric computation in shaping feasible approximations and runtimes.

Abstract

The $k$-center problem is a central optimization problem with numerous applications for machine learning, data mining, and communication networks. Despite extensive study in various scenarios, it surprisingly has not been thoroughly explored in the traditional distributed setting, where the communication graph of a network also defines the distance metric. We initiate the study of the $k$-center problem in a setting where the underlying metric is the graph's shortest path metric in three canonical distributed settings: the LOCAL, CONGEST, and CLIQUE models. Our results encompass constant-factor approximation algorithms and lower bounds in these models, as well as hardness results for the bi-criteria approximation setting.

$k$-Center Clustering in Distributed Models

TL;DR

This work initiates the study of k-center in distributed models where the network graph defines the metric via shortest paths, examining LOCAL, CONGEST, and CLIQUE settings. It develops a spectrum of results: a simple (2+ε)k-approximation in the LOCAL model and a tight linear-time barrier for better-than-(k−1) performance, a 2-approximation in CONGEST using BFS-based greedy techniques, and a near-all-pairs-shortest-path–driven 2-approximation in the CLIQUE model with varying trade-offs dependent on distance computation accuracy. The paper also establishes strong lower bounds in CONGEST via reductions from disjointness, extended to k-center, and discusses the challenges of lower bounds in the CLIQUE model. Collectively, the results delineate the distributed complexity landscape of graph-metric k-center, highlighting the role of communication constraints and metric computation in shaping feasible approximations and runtimes.

Abstract

The -center problem is a central optimization problem with numerous applications for machine learning, data mining, and communication networks. Despite extensive study in various scenarios, it surprisingly has not been thoroughly explored in the traditional distributed setting, where the communication graph of a network also defines the distance metric. We initiate the study of the -center problem in a setting where the underlying metric is the graph's shortest path metric in three canonical distributed settings: the LOCAL, CONGEST, and CLIQUE models. Our results encompass constant-factor approximation algorithms and lower bounds in these models, as well as hardness results for the bi-criteria approximation setting.
Paper Structure (19 sections, 9 theorems, 4 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 19 sections, 9 theorems, 4 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Distributed $k$-Center
  3. Our Results and Techniques
  4. Preliminaries
  5. The $k$-Center Problem
  6. Computational Models
  7. Communication Complexity
  8. Related Work
  9. $k$-Center in Related Computational Models
  10. Metric Facility Location
  11. The congest Model
  12. The clique Model
  13. Distributed Large-Scale Computational Models
  14. The $k$-Center Problem in the local Model
  15. A $2$-Approximation in the congest Model
  16. ...and 4 more sections

Key Result

Lemma 1

For any $\epsilon>0$ there is a deterministic $O(k/\epsilon)$-round algorithm in the local model that gives a $((2+\epsilon)k)$-approximate solution for the $k$-center problem.

Figures (2)

  • Figure 1: Illustration for the proof of Theorem \ref{['thm:lower:bound:local']}. Left: cycle $C$. Right: cycle $C'$
  • Figure 2: Illustration of the graph $G_{x, y}$AbboudCKP21. The dotted edges depend on the inputs for the disjointness problem.

Theorems & Definitions (18)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2: DBLP:journals/tcs/Gonzalez85
  • Theorem 2
  • Claim 1: AbboudCKP21
  • Lemma 3
  • proof
  • Theorem 3
  • ...and 8 more