The Lasserre Rank of the Cropped Hypercube

Abstract

In an $n$-dimensional \emph{cropped hypercube} each of the $2^n$ cropping inequalities chops off a single corner of the $0$--$1$ hypercube by an $\ell_1$-distance $ρ$. The case $ρ= 1/2$ has been extensively studied in the literature. This paper shows that the Lasserre rank of the $n$-dimensional cropped hypercube where $ρ= 1/2$, $n \geq 2$, is the smallest integer $0\leq t \leq n$ such that $Δ_t < 0$ in the recurrence $Δ_{-1} = 1$, $Δ_{0} = n-1$, $Δ_t = (n-1)Δ_{t-1} - t(n-t+1)Δ_{t-2}$. It follows that the Lasserre rank can be computed in time $O(n^2 \log^2 n)$. Asymptotically, the rank is $\frac{n}{2} + c_{1/2}\sqrt{n} + o(\sqrt{n})$, where $c_{1/2}$ is the unique zero of a given function. Numerically, $c_{1/2} \approx 0.3825$. In fact, we prove such results for any fixed $0 < ρ< 1$.