  An instance of colorful k-center consists of points in a metric space that
are colored red or blue, along with an integer k and a coverage requirement for
each color. The goal is to find the smallest radius \r{ho} such that there
exist balls of radius \r{ho} around k of the points that meet the coverage
requirements. The motivation behind this problem is twofold. First, from
fairness considerations: each color/group should receive a similar service
guarantee, and second, from the algorithmic challenges it poses: this problem
combines the difficulties of clustering along with the subset-sum problem. In
particular, we show that this combination results in strong integrality gap
lower bounds for several natural linear programming relaxations. Our main
result is an efficient approximation algorithm that overcomes these
difficulties to achieve an approximation guarantee of 3, nearly matching the
tight approximation guarantee of 2 for the classical k-center problem which
this problem generalizes.
