On Permuting Some Coordinates of Polytopes

Abstract

Let $P\subseteq \mathbb {R}^{d}$ be a polytope with coordinates labeled $x_1,\ldots ,x_d$. Define $\textrm{perm}_I(P)$ to be the polytope obtained by taking every permutation $\sigma$ whose set of fixed-points is $[d]\setminus I$, permuting the coordinates of every point in P according to $\sigma$ and taking the convex hull of all such points. Also, define $\textrm{sort}(P)$ to be the polytope obtained by taking each vertex of P in “sorted order".

In this article we study the extension complexity of $\textrm{perm}_I(P)$ and $\textrm{sort}(P)$ in terms of the extension complexity of P. A result by Kaibel and Pashkovich states that if $\textrm{sort}(P)\subseteq P$ and $I=[d]$ then the extension complexity of $\textrm{perm}_I(P)$ is bounded above by a polynomial of the extension complexity of P. We show that the extension complexity of $\textrm{perm}_I(P)$ can increase exponentially if $I\ne [d]$ even if the vertices of P contain only three values, say 0, 1,  or 2 at each of the coordinates $x_i$ for $i\in I$. Furthermore, the extension complexity of $\textrm{sort}(P)$ can be exponentially larger than that of P. We also discuss the implications for the 0/1 case.