On the Vertices of Delta-modular Polyhedra
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2025
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| author | Mikhail, Bludov Dmitry, Gribanov Maxim, Klimenko Andrey, Kupavskii Zsolt, Lángi Alexander, Rogozin Vsevolod, Voronov |
| author_facet | Mikhail, Bludov Dmitry, Gribanov Maxim, Klimenko Andrey, Kupavskii Zsolt, Lángi Alexander, Rogozin Vsevolod, Voronov |
| contents | 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 $P$: $$
n! \cdot \fracΔ{Δ_{\text{average}}} \cdot \text{vol}(B_2) \sim \frac{1}{\sqrt{πn}} \cdot \left(\frac{2 π}{e}\right)^{n/2} \cdot n^{n/2} \cdot \fracΔ{Δ_{\text{average}}}, $$ where $Δ$ is the maximum absolute value of $n \times n$ subdeterminants of $A$, and $Δ_{\text{average}}$ 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 $Δ$-modular polyhedra. Note that in the integer case, the bound can be simplified via the inequality $Δ_{\text{average}} \geq Δ_{\min} \geq 1$, where $Δ_{\min}$ 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 $O\bigl(n^3 \cdot \fracΔ{Δ_{\min}} \cdot \ln (n \fracΔ{Δ_{\min}}) \bigr)$. Thus, both characteristic of $P$ are linear in $Δ/Δ_{\min}$.
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 $O(n)^{n/2} \cdot m^2 \cdot \fracΔ{Δ_{\text{average}}}$ 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 $O(n)^n \cdot \frac{Δ^4}{Δ_{\text{average}}}$ arithmetic operations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Vertices of Delta-modular Polyhedra Mikhail, Bludov Dmitry, Gribanov Maxim, Klimenko Andrey, Kupavskii Zsolt, Lángi Alexander, Rogozin Vsevolod, Voronov Combinatorics Computational Geometry Discrete Mathematics 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 $P$: $$ n! \cdot \fracΔ{Δ_{\text{average}}} \cdot \text{vol}(B_2) \sim \frac{1}{\sqrt{πn}} \cdot \left(\frac{2 π}{e}\right)^{n/2} \cdot n^{n/2} \cdot \fracΔ{Δ_{\text{average}}}, $$ where $Δ$ is the maximum absolute value of $n \times n$ subdeterminants of $A$, and $Δ_{\text{average}}$ 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 $Δ$-modular polyhedra. Note that in the integer case, the bound can be simplified via the inequality $Δ_{\text{average}} \geq Δ_{\min} \geq 1$, where $Δ_{\min}$ 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 $O\bigl(n^3 \cdot \fracΔ{Δ_{\min}} \cdot \ln (n \fracΔ{Δ_{\min}}) \bigr)$. Thus, both characteristic of $P$ are linear in $Δ/Δ_{\min}$. 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 $O(n)^{n/2} \cdot m^2 \cdot \fracΔ{Δ_{\text{average}}}$ 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 $O(n)^n \cdot \frac{Δ^4}{Δ_{\text{average}}}$ arithmetic operations. |
| title | On the Vertices of Delta-modular Polyhedra |
| topic | Combinatorics Computational Geometry Discrete Mathematics |
| url | https://arxiv.org/abs/2506.12774 |