# On connections between k-coloring and Euclidean k-means

arXiv: https://arxiv.org/abs/2405.13877 · HTML: https://mlsanity.com/paper/2405.13877/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2024.

## Built on by (most cited first, 1 of 1)

- [On Approximability of ℓ22 Min-Sum Clustering](https://mlsanity.com/paper/2412.03332/) (2024)

## Abstract (arXiv)

In the Euclidean k-means problems we are given as input a set of n points in ℝd and the goal is to find a set of k points Csubseteq ℝd, 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: bullet 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. bullet 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. bullet We prove similar results and connections as above for the Euclidean k-min-sum problem.