Asymmetric kappa-center is log* n-hard to approximate
In the ASYMMETRIC k-CENTER problem, the input is an integer k and a complete digraph over n points together with a distance function obeying the directed triangle inequality. The goal is to choose a set of k points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point from its center is as small as possible. We show that the ASYMMETRIC k-CENTER problem is hard to approximate up to a factor of log* n -O(i) unless NP subset of DTIME(n(log log n)). Since an O(log*n) -approximation algorithm is known for this problems this resolves the asymptotic approximability of,AsYMMETRic k-CENTER. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric (symmetric) k-Center problem with costs.
asymmetric kappa-center algorithms theory approximation algorithms hardness of approximation metric kappa-center