  This paper studies the scaling of the expected total queue size in an
$n\times n$ input-queued switch, as a function of both the load $\rho$ and the
system scale $n$. We provide a new class of scheduling policies under which the
expected total queue size scales as $O\left( n(1-\rho)^{-4/3} \log
\left(\max\{\frac{1}{1-\rho}, n\}\right)\right)$, over all $n$ and $\rho<1$,
when the arrival rates are uniform. This improves over the previously
best-known scalings in two regimes: $O\left(n^{1.5}(1-\rho)^{-1} \log
\frac{1}{1-\rho}\right)$ when $\Omega(n^{-1.5}) \le 1-\rho \le O(n^{-1})$ and
$O\left(\frac{n\log n}{(1-\rho)^2}\right)$ when $1-\rho \geq \Omega(n^{-1})$. A
key ingredient in our method is a tight characterization of the largest
$k$-factor of a random bipartite multigraph, which may be of independent
interest.
