Low degree sum-of-squares bounds for the stability number: a copositive approach

Abstract

The stability number of a graph $G$, denoted as $\alpha(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lov\'asz's theta number and expanded through lift-and-project hierarchies as well as sums of squares (SOS) relaxations, provide powerful tools for approximating $\alpha(G)$. We build upon the copositive formulation of $\alpha(G)$ and introduce a novel SDP-based hierarchy of inner approximations to the copositive cone COP$_n$, which is derived from structured SOS representations. This hierarchy preserves essential structural properties that are missing in existing approaches, offers an SDP feasibility formulation at each level despite its non-convexity, and converges finitely to $\alpha(G)$. Our results include examples of graph families that require at least $\alpha(G) - 1$ levels for related hierarchies, indicating the tightness of the de Klerk-Pasechnik conjecture. Notably, on those graph families, our hierarchy achieves $\alpha(G)$ in a single step.