On the Vertices of Delta-modular Polyhedra
Let $P$ be a polytope defined by the system $A x \leq b$, where $A \in R^{m \times n}$, $b \in R^m$, and $\text{rank}(A) = n$. We give a short geometric proof of the following tight upper bound on the number of vertices of $P$: $$ n! \cdot \frac{\Delta}{\Delta_{\text{average}}} \cdot \text{vol}(B_2) \sim \frac{1}{\sqrt{\pi n}} \cdot \left(\frac{2 \pi}{e}\right)^{n/2} \cdot n^{n/2} \cdot \frac{\Delta}{\Delta_{\text{average}}}, $$ where $\Delta$ is the maximum absolute value of $n \times n$ subdeterminants of $A$, and $\Delta_{\text{average}}$ is the average absolute value of subdeterminants of $A$ corresponding to a triangulation of $P$'s normal fan. Assuming that $A$ is integer, such polyhedra are called $\Delta$-modular polyhedra. Note that in the integer case, the bound can be simplified via the inequality $\Delta_{\text{average}} \geq \Delta_{\min} \geq 1$, where $\Delta_{\min}$ is the minimum absolute value of subdeterminants of $A$ corresponding to feasible bases of $A x \leq b$. For this, we prove and use a symmetric variant of Macbeath's theorem. Additionally, we give a direct argument based on prior results in the field, showing that the graph diameter of $P$ is bounded by $O\bigl(n^3 \cdot \frac{\Delta}{\Delta_{\min}} \cdot \ln (n \frac{\Delta}{\Delta_{\min}}) \bigr)$. Thus, both characteristic of $P$ are linear in $\Delta/\Delta_{\min}$. From an algorithmic perspective, we demonstrate that: Given $A \in Q^{m \times n}$, $b \in Q^m$, and an initial feasible solution to $A x \leq b$, the convex hull of $P$ can be constructed in $O(n)^{n/2} \cdot m^2 \cdot \frac{\Delta}{\Delta_{\text{average}}}$ operations. For simple polyhedra, the dependence on $m$ reduces to linear; Given $A \in Z^{m \times n}$ and $b \in Q^m$, the number $|P \cap Z^n|$ can be computed in $O(n)^n \cdot \frac{\Delta^4}{\Delta_{\text{average}}}$ arithmetic operations.