  For a finite set of balls of radius $r$, the $k$-fold cover is the space
covered by at least $k$ balls. Fixing the ball centers and varying the radius,
we obtain a nested sequence of spaces that is called the $k$-fold filtration of
the centers. For $k=1$, the construction is the union-of-balls filtration that
is popular in topological data analysis. For larger $k$, it yields a cleaner
shape reconstruction in the presence of outliers. We contribute a
sparsification algorithm to approximate the topology of the $k$-fold
filtration. Our method is a combination and adaptation of several techniques
from the well-studied case $k=1$, resulting in a sparsification of linear size
that can be computed in expected near-linear time with respect to the number of
input points. Our method also extends to the multicover bifiltration, composed
of the $k$-fold filtrations for several values of $k$, with the same size and
complexity bounds.
