  Fair clustering enjoyed a surge of interest recently. One appealing way of
integrating fairness aspects into classical clustering problems is by
introducing multiple covering constraints. This is a natural generalization of
the robust (or outlier) setting, which has been studied extensively and is
amenable to a variety of classic algorithmic techniques. In contrast, for the
case of multiple covering constraints (the so-called colorful setting),
specialized techniques have only been developed recently for $k$-Center
clustering variants, which is also the focus of this paper. While prior
techniques assume covering constraints on the clients, they do not address
additional constraints on the facilities, which has been extensively studied in
non-colorful settings.
  In this paper, we present a quite versatile framework to deal with various
constraints on the facilities in the colorful setting, by combining ideas from
the iterative greedy procedure for Colorful $k$-Center by Inamdar and
Varadarajan with new ingredients. To exemplify our framework, we show how it
leads, for a constant number $\gamma$ of colors, to the first constant-factor
approximations for both Colorful Matroid Supplier with respect to a linear
matroid and Colorful Knapsack Supplier. In both cases, we readily get an
$O(2^\gamma)$-approximation.
  Moreover, for Colorful Knapsack Supplier, we show that it is possible to
obtain constant approximation guarantees that are independent of the number of
colors $\gamma$, as long as $\gamma=O(1)$, which is needed to obtain a
polynomial running time. More precisely, we obtain a $7$-approximation by
extending a technique recently introduced by Jia, Sheth, and Svensson for
Colorful $k$-Center.
