Extremal integrality ratios for fixed graph order and LS_+ level

Characterize, for every pair of positive integers (n,ℓ) with n≥ℓ, the n-vertex graphs that maximize the integrality ratio $\alpha_{\LS_+^{\ell}(G)}/\alpha(G)$, where $\alpha_{\LS_+^{\ell}(G)}=\max\{\bar e^{\top}x:x\in\LS_+^{\ell}(G)\}$.

Background

The paper studies LS_+-rank as a measure of how many hierarchy iterations are required to recover the stable set polytope. It identifies ℓ-minimal graphs as worst-case examples for rank, but notes that integrality ratios provide a different measure of relaxation quality. For fixed numbers of vertices and LS_+ iterations, the extremal graph structures maximizing this ratio are not characterized. The problem asks for a complete classification across all admissible pairs (n,ℓ).

References

For each pair of positive integers $(n, \ell)$ with $n \geq \ell$, characterize the family of graphs $G$ on $n$ vertices which maximize the integrality ratio:

\frac{\alpha_{\LS_+{\ell}(G)}{\alpha(G)},

— Stable Set Polytopes with Rank $|V(G)|/3$ for the Lovász--Schrijver SDP Operator  (2501.07413 - Au et al., 13 Jan 2025) in Section 6, third displayed Open Problem