{"ID":2874175,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.04949","arxiv_id":"2509.04949","title":"Low degree sum-of-squares bounds for the stability number: a copositive approach","abstract":"The stability number of a graph $G$, denoted as $α(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lovász's theta number and expanded through lift-and-project hierarchies as well as sums of squares (SOS) relaxations, provide powerful tools for approximating $α(G)$. We build upon the copositive formulation of $α(G)$ 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 $α(G)$. Our results include examples of graph families that require at least $α(G) - 1$ levels for related hierarchies, indicating the tightness of the de Klerk-Pasechnik conjecture. Notably, on those graph families, our hierarchy achieves $α(G)$ in a single step.","short_abstract":"The stability number of a graph $G$, denoted as $α(G)$, is the maximum size of an independent (stable) set in $G$. Semidefinite programming (SDP) methods, which originated from Lovász's theta number and expanded through lift-and-project hierarchies as well as sums of squares (SOS) relaxations, provide powerful tools fo...","url_abs":"https://arxiv.org/abs/2509.04949","url_pdf":"https://arxiv.org/pdf/2509.04949v2","authors":"[\"Luis Felipe Vargas\",\"Juan C. Vera\",\"Peter J. C. Dickinson\"]","published":"2025-09-05T09:15:09Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"math.CO\"]","methods":"[]","has_code":false}