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$n \ge d \ge 1$.