On Connections Between k-Coloring and Euclidean k-Means
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On Connections Between k-Coloring and Euclidean k-Means

Authors Enver Aman, Karthik C. S. , Sharath Punna

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Author Details

Enver Aman
  • Rutgers University, Piscataway, NJ, USA
Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
Sharath Punna
  • Rutgers University, Piscataway, NJ, USA

Funding

This work was supported by a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen and by the National Science Foundation under Grant CCF-2313372.

Acknowledgements

We would like to thank Pasin Manurangsi for pointing us to [Björklund et al., 2007] and informing us that the fast max-sum convolution result of that paper can be used to obtain a 2ⁿ⋅ poly(n,d) runtime algorithm for the k-means problem. Also, we would like to thank Vincent Cohen-Addad for suggesting to us the that 2-min-sum problem might be more naturally connected to the Max-Cut problem. Finally, we would like to thank the anonymous reviewers for helping us improve the presentation of the paper.

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Enver Aman, Karthik C. S., and Sharath Punna. On Connections Between k-Coloring and Euclidean k-Means. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.9

Abstract

In the Euclidean k-means problems we are given as input a set of n points in ℝ^d and the goal is to find a set of k points C ⊆ ℝ^d, so as to minimize the sum of the squared Euclidean distances from each point in P to its closest center in C. In this paper, we formally explore connections between the k-coloring problem on graphs and the Euclidean k-means problem. Our results are as follows:  
- For all k ≥ 3, we provide a simple reduction from the k-coloring problem on regular graphs to the Euclidean k-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean 2-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem. 
- In the other direction, we mimic the O(1.7297ⁿ) time algorithm of Williams [TCS'05] for the max-cut of problem on n vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in 2ⁿ⋅ poly(n,d) time. 
- We prove similar results and connections as above for the Euclidean k-min-sum problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity classes
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • k-means
  • k-minsum
  • Euclidean space
  • fine-grained complexity

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