Hardness of Approximation for Euclidean k-Median

The Euclidean k-median problem is defined in the following manner: given a set 𝒳 of n points in d-dimensional Euclidean space ℝ^d, and an integer k, find a set C ⊂ ℝ^d of k points (called centers) such that the cost function Φ(C,𝒳) ≡ ∑_{x ∈ 𝒳} min_{c ∈ C} ‖x-c‖₂ is minimized. The Euclidean k-means problem is defined similarly by replacing the distance with squared Euclidean distance in the cost function. Various hardness of approximation results are known for the Euclidean k-means problem [Pranjal Awasthi et al., 2015; Euiwoong Lee et al., 2017; Vincent Cohen{-}Addad and {Karthik {C. S.}}, 2019]. However, no hardness of approximation result was known for the Euclidean k-median problem. In this work, assuming the unique games conjecture (UGC), we provide the hardness of approximation result for the Euclidean k-median problem in O(log k) dimensional space. This solves an open question posed explicitly in the work of Awasthi et al. [Pranjal Awasthi et al., 2015]. Furthermore, we study the hardness of approximation for the Euclidean k-means/k-median problems in the bi-criteria setting where an algorithm is allowed to choose more than k centers. That is, bi-criteria approximation algorithms are allowed to output β k centers (for constant β > 1) and the approximation ratio is computed with respect to the optimal k-means/k-median cost. We show the hardness of bi-criteria approximation result for the Euclidean k-median problem for any β < 1.015, assuming UGC. We also show a similar hardness of bi-criteria approximation result for the Euclidean k-means problem with a stronger bound of β < 1.28, again assuming UGC.