On a Simple Connection Between $\Delta$-modular ILP and LP, and a New Bound on the Number of Integer Vertices

TL;DR

This paper establishes a connection between $ ext{Delta}$-modular integer linear programming and linear programming, providing a new bound on the number of integer vertices of the associated polyhedron that improves previous bounds for certain $ ext{Delta}$ values.

Contribution

It generalizes a known geometric fact to $ ext{Delta}$-modular polyhedra and derives a tighter upper bound on the number of integer vertices, enhancing understanding of $ ext{Delta}$-modular ILPs.

Findings

Proves a $ ext{Delta}$-approximate version of the face-inequality property for integer polyhedra.

Derives an upper bound of $2 inom{m}{n} ext{Delta}^{n-1}$ on the number of integer vertices.

Improves existing bounds for the number of vertices when $ ext{Delta} = O(n^2)$.

Abstract

Let $A \in Z^{m \times n}$, $rank(A) = n$, $b \in Z^m$, and $P$ be an $n$-dimensional polyhedron, induced by the system $A x \leq b$. It is a known fact that if $F$ is a $k$-face of $P$, then there exist at least $n-k$ linearly independent inequalities of the system $A x \leq b$ that become equalities on $F$. In other words, there exists a set of indices $J$, such that $|J| \geq n-k$, $rank(A_{J}) = n-k$, and $$ A_{J} x - b_{J} = 0,\quad \text{for any $x \in F$}. $$ We show that a similar fact holds for the integer polyhedron $$ P_{I} = conv.hull\bigl(P \cap Z^n\bigr), $$ if we additionally suppose that $P$ is $\Delta$-modular, for some $\Delta \in \{1,2,\dots\}$. More precisely, if $F$ is a $k$-face of $P_{I}$, then there exists a set of indices $J$, such that $|J| \geq n-k$, $rank(A_{J}) = n-k$, and $$ A_{J} x - b_{J} \overset{\Delta}{=} 0,\quad \text{for any $x \in F \cap Z^n$}, $$ where $x \overset{\Delta}{=} y$ means that $\|x - y\|_{\infty} < \Delta$. In other words, there exist at least $n-k$ linearly independent inequalities of the system $A x \leq b$ that almost become equalities on $F \cap Z^n$. When we say almost, we mean that the slacks are not greater than $\Delta-1$. Using this fact, we prove the inequality $$ |vert(P_I)| \leq 2 \cdot \binom{m}{n} \cdot \Delta^{n-1}, $$ for the number of vertices of $P_I$, which is better, than the state of the art bound for $\Delta = O(n^2)$.