  We study core-set construction algorithms for the task of Diversity
Maximization under fairness/partition constraint. Given a set of points $P$ in
a metric space partitioned into $m$ groups, and given $k_1,\ldots,k_m$, the
goal of this problem is to pick $k_i$ points from each group $i$ such that the
overall diversity of the $k=\sum_i k_i$ picked points is maximized. We consider
two natural diversity measures: sum-of-pairwise distances and
sum-of-nearest-neighbor distances, and show improved core-set construction
algorithms with respect to these measures. More precisely, we show the first
constant factor core-set w.r.t. sum-of-pairwise distances whose size is
independent of the size of the dataset and the aspect ratio. Second, we show
the first core-set w.r.t. the sum-of-nearest-neighbor distances. Finally, we
run several experiments showing the effectiveness of our core-set approach. In
particular, we apply constrained diversity maximization to summarize a set of
timed messages that takes into account the messages' recency. Specifically, the
summary should include more recent messages compared to older ones. This is a
real task in one of the largest communication platforms, affecting the
experience of hundreds of millions daily active users. By utilizing our
core-set method for this task, we achieve a 100x speed-up while losing the
diversity by only a few percent. Moreover, our approach allows us to improve
the space usage of the algorithm in the streaming setting.
