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

TL;DR

It is shown that any bounded integral function $f$ with rank $r$ has deterministic communication complexity $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log r$ and 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

Abstract

We show that any bounded integral function $f : A \times B \mapsto \{0,1, \dots, Δ\}$ with rank $r$ has deterministic communication complexity $Δ^{O(Δ)} \cdot \sqrt{r} \cdot \log r$, 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 $\{0,1,\dots,Δ\}$ has extension complexity at most $\exp(Δ^{O(Δ)} \cdot \sqrt{n} \cdot \log n)$.

Theorems & Definitions (26)

• Theorem 1: Main result, communication complexity
• Corollary 2: Main result, extension complexity
• proof
• Lemma 3
• proof
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof