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Polytopes with Bounded Integral Slack Matrices Have Sub-Exponential Extension Complexity (2307.16159v3)
Published 30 Jul 2023 in cs.DM and math.CO
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 $\exp(\Delta{O(\Delta)} \cdot \sqrt{n} \cdot \log n)$.