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

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 $\exp(\Delta^{O(\Delta)} \cdot \sqrt{n} \cdot \log n)$.