A (4+ϵ)-Approximation for Euclidean k-Means via Non-monotone Dual-Fitting

We present a polynomial-time (4+є)-approximation algorithm for (high-dimensional) Euclidean k-Means. This substantially improves on the current-best 5.83-approximation in [Charikar, Cohen-Addad, Gao, Grandoni, Lee, Van Wijland - FOCS’25] (that also works for the metric case). The mentioned algorithm by Charikar et al. critically exploits a greedy Lagrangian Multiplier Preserving (LMP) approximation for Facility Location with squared metric distances, that adapts the classical greedy algorithm with dual-fitting analysis for Metric Facility Location in [Jain, Mahdian, Markakis, Saberi, Vazirani - J.ACM’03]. The authors then turn it into an approximation algorithm for (Metric) k-Means, at the cost on an extra factor 1+є, by exploiting the framework introduced in [Cohen-Addad, Grandoni, Lee, Schwiegelshohn, Svensson - STOC’25] for k-Median. Our main contribution is a greedy LMP 4-approximation for Facility Location with squared Euclidean distances. Differently from Charikar et al., our algorithm sometimes decreases the dual variables, a quite uncommon feature for dual-based algorithms. This is critical in our dual-fitting analysis in order to exploit the specific properties of Euclidean metrics. For the (4+є)-approximation for k-Means, we extend the framework by Cohen-Addad et al. by overcoming substantial technical challenges posed by decreased dual values.