  We study a variant of the canonical k-center problem over a set of vertices
in a metric space, where the underlying distances are apriori unknown. Instead,
we can query an oracle which provides noisy/incomplete estimates of the
distance between any pair of vertices. We consider two oracle models: Dimension
Sampling where each query to the oracle returns the distance between a pair of
points in one dimension; and Noisy Distance Sampling where the oracle returns
the true distance corrupted by noise. We propose active algorithms, based on
ideas such as UCB, Thompson Sampling and Track-and-Stop developed in the
closely related Multi-Armed Bandit problem, which adaptively decide which
queries to send to the oracle and are able to solve the k-center problem within
an approximation ratio of two with high probability. We analytically
characterize instance-dependent query complexity of our algorithms and also
demonstrate significant improvements over naive implementations via numerical
evaluations on two real-world datasets (Tiny ImageNet and UT Zappos50K).
