Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope

TL;DR

This paper introduces a new geometric graph parameter, gd(G), characterizing low-rank solutions to semidefinite programs over the elliptope, with a complete minor characterization for the case r=2 and bounds related to a tree-width-like parameter.

Contribution

The paper defines gd(G), proves its minor monotonicity, characterizes forbidden minors for gd(G) r, and relates it to a new tree-width-like parameter la(G), advancing understanding of low-rank semidefinite solutions.

Findings

gd(G) is minor monotone.

Complete forbidden minor characterization for gd(G) 2.

gd(G)  2 iff la(G)  2 for certain graphs.

Abstract

We study a new geometric graph parameter $\egd(G)$, defined as the smallest integer $r\ge 1$ for which any partial symmetric matrix which is completable to a correlation matrix and whose entries are specified at the positions of the edges of $G$, can be completed to a matrix in the convex hull of correlation matrices of $\rank $ at most $r$. This graph parameter is motivated by its relevance to the problem of finding low rank solutions to semidefinite programs over the elliptope, and also by its relevance to the bounded rank Grothendieck constant. Indeed, $\egd(G)\le r$ if and only if the rank-$r$ Grothendieck constant of $G$ is equal to 1. We show that the parameter $\egd(G)$ is minor monotone, we identify several classes of forbidden minors for $\egd(G)\le r$ and we give the full characterization for the case $r=2$. We also show an upper bound for $\egd(G)$ in terms of a new tree-width-like parameter $\sla(G)$, defined as the smallest $r$ for which $G$ is a minor of the strong product of a tree and $K_r$. We show that, for any 2-connected graph $G\ne K_{3,3}$ on at least 6 nodes, $\egd(G)\le 2$ if and only if $\sla(G)\le 2$.