k-Center Clustering Under Perturbation Resilience

The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric k-center and an O(log∗(k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case. In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a very natural input stability (promise) condition called αperturbation resilience [11], which states that the optimal solution does not change under any α-factor perturbation to the input distances. We show that by assuming 2-perturbation resilience, the exact solution for the asymmetric k-centerproblemcanbefoundinpolynomialtime.Toourknowledge,thisisthe first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solvedinpolynomialtimeunderperturbationresilienceforaconstantvalueof α. Furthermore, we prove our result is tight by showing symmetric k-center under (2 − )-perturbation resilience is hard unless NP = RP.Thisisthefirsttightresultforanyproblemunderperturbationresilience,i.e.,thisisthefirst time the exact value of α for which the problem switches from being NP-hard to efficiently computable has been found. Ourresultsillustrateasurprisingrelationshipbetweensymmetricandasymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience to 2-perturbations.