On connections between k-coloring and Euclidean k-means

TL;DR

This work builds a bridge between graph coloring and Euclidean clustering, delivering reductions that transfer hardness and algorithmic ideas between $k$-coloring, Balanced Max-Cut, and Euclidean $k$-means (as well as $k$-min-sum). It provides a simple linear-size reduction from $k$-coloring on $d$-regular graphs to Euclidean $k$-means for any fixed $k\ge 3$, yielding an alternate NP-hardness proof for $k$-means, and extends to a structured max-cut variant to Euclidean $2$-means. On the algorithmic side, the paper presents an exact $1.7297^n\cdot \mathrm{poly}(n,d)$ time algorithm for 2-means by reducing to Weighted 2-CSP and employing fast matrix-multiplication-based CSP solvers, with a conditional improvement to $1.59^n$ under $\omega=2$. A parallel fine-grained analysis shows a computational equivalence between 2-min-sum and Balanced Max-Cut in $\ell_p$-metrics, and proves a matching $1.7297^n$-time algorithm for 2-min-sum via Max-Cut reductions. Together, these results illuminate the transfer of complexity and technique between combinatorial graph problems and geometric clustering, and open avenues for leveraging colorability tools to accelerate clustering problems. The findings have potential implications for both hardness proofs and practical algorithm design in high-dimensional clustering and related optimization tasks.

Abstract

In the Euclidean $k$-means problems we are given as input a set of $n$ points in $\mathbb{R}^d$ and the goal is to find a set of $k$ points $C\subseteq \mathbb{R}^d$, 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: $\bullet$ For all $k\ge 3$, 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 $O(1.7297^n)$ 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 $2^n\cdot \text{poly}(n,d)$ time. $\bullet$ We prove similar results and connections as above for the Euclidean $k$-min-sum problem.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 2.1: dailey1980uniqueness
• Theorem 2.2
• Claim 3.1
• proof
• Remark 3.2
• Theorem 5.1
• Corollary 5.2
• proof
• Theorem 5.3: Theorem 6.4.1 in williams2007algorithms