Improving bounds on the diameter of a polyhedron in high dimensions

In 1992, Kalai and Kleitman proved that the diameter of a $d$-dimensional polyhedron with $n$ facets is at most $n^{2+\log_2 d}$. In 2014, Todd improved the Kalai-Kleitman bound to $(n-d)^{\log_2 d}$. We improve the Todd bound to $(n-d)^{-1+\log_2 d}$ for $n \ge d \ge 7$, $(n-d)^{-2+\log_2 d}$ for $n \ge d \ge 37$, and $(n-d)^{-3+\log_2 d+O\left(1/d\right)}$ for $n \ge d \ge 1$.