Polytopes with Bounded Integral Slack Matrices Have Sub-Exponential Extension Complexity

TL;DR

This paper establishes a connection between the bounded integral slack matrices of polytopes and their extension complexity, showing that such polytopes have sub-exponential extension complexity based on their dimension.

Contribution

It introduces bounds on the extension complexity of polytopes with bounded integral slack matrices, linking combinatorial properties to geometric complexity.

Findings

Bounded integral functions have communication complexity depending on rank and maximum value.

Polytopes with bounded integral slack matrices have sub-exponential extension complexity.

Extension complexity is at most exponential in a function of the polytope's dimension.

Abstract

We show that any bounded integral function $f : A \times B \mapsto \{0,1, \dots, \Delta\}$ with rank $r$ has deterministic communication complexity $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log r$, where the rank of $f$ is defined to be the rank of the $A \times B$ matrix whose entries are the function values. As a corollary, we show that any $n$-dimensional polytope that admits a slack matrix with entries from $\{0,1,\dots,\Delta\}$ has extension complexity at most $\exp(\Delta^{O(\Delta)} \cdot \sqrt{n} \cdot \log n)$.