Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants - Bandyapadhyay, Sayan; Friggstad, Zachary; Mousavi, Ramin (Algorithmica, 2024)
k -center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an O ( n log k ) + ( 1 / ϵ ) O ( 2 d k 1 - 1 / d log k ) -time ( 1 + ϵ ) -approximation for Euclidean k -center, where d is the dimension. We show for a closely related problem, k -supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k -center and k -supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k -center, called Non-uniform k -center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2 O ( k log k ) n 2 time 3-approximation for NUkC in general metrics, and a 2 O ( ( k log k ) / ϵ ) d n time ( 1 + ϵ ) -approximation for Euclidean NUkC. The latter time bound matches the bound for k -center.