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

07/30/2023

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by   Sally Dong, et al.

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We show that any bounded integral function f : A × B ↦{0,1, …, Δ} with rank r has deterministic communication complexity Δ^O(Δ)·√(r)·log^2 r, where the rank of f is defined to be the rank of the A × 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,…,Δ} has extension complexity at most exp(Δ^O(Δ)·√(n)·log^2 n).