  Given a finite set $A\subset\mathbb{R}^d$, let Cov$_{r,k}$ denote the set of
all points within distance $r$ to at least $k$ points of $A$. Allowing $r$ and
$k$ to vary, we obtain a 2-parameter family of spaces that grow larger when $r$
increases or $k$ decreases, called the \emph{multicover bifiltration}.
Motivated by the problem of computing the homology of this bifiltration, we
introduce two closely related combinatorial bifiltrations, one polyhedral and
the other simplicial, which are both topologically equivalent to the multicover
bifiltration and far smaller than a \v Cech-based model considered in prior
work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid
tiling of Edelsbrunner and Osang, and can be efficiently computed using a
variant of an algorithm given by these authors. Using an implementation for
dimension 2 and 3, we provide experimental results. Our simplicial construction
is useful for understanding the polyhedral construction and proving its
correctness.
