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In the Euclidean k-Means problem we are given a collection of n points D in an Euclidean space and a positive integer k. Our goal is to identify a collection of k points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in D and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual 6.357 approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20]. In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved 6.12903 approximation. As a related result, we also show that the mentioned LP has integrality gap at least (16+Sqrt(5))/15 > 1.2157. .more » « less
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Rabani, Yuval ; Schulman, Leonard J. ( , Journal of Mathematical Economics)null (Ed.)
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Dvijotham, Krishnamurthy ; Rabani, Yuval ; Schulman, Leonard J. ( , Games and Economic Behavior)null (Ed.)