On connections between k-coloring and Euclidean k-means

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 C 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: 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.