Improved approximation algorithms for solving the squared metric k-facility location problem

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The squared metric k-facility location problem is a frequently encountered generalization of the k-means problem, where a specific cost should be paid for opening each facility. The current best approximation ratio for this problem is 44.473+ϵ, which was obtained using a local search algorithm. We advance the state-of-the-art for the problem by devising a Lagrangian relaxation-based algorithm that achieves an improved approximation guarantee of 36.342+ϵ. Our improvement comes from a new deterministic rounding approach, which exploits the properties of the squared metric.