Low degree sum-of-squares bounds for the stability number: a copositive approach
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914031122513920 |
|---|---|
| author | Vargas, Luis Felipe Vera, Juan C. Dickinson, Peter J. C. |
| author_facet | Vargas, Luis Felipe Vera, Juan C. Dickinson, Peter J. C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_04949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low degree sum-of-squares bounds for the stability number: a copositive approach Vargas, Luis Felipe Vera, Juan C. Dickinson, Peter J. C. Optimization and Control Combinatorics 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. |
| title | Low degree sum-of-squares bounds for the stability number: a copositive approach |
| topic | Optimization and Control Combinatorics |
| url | https://arxiv.org/abs/2509.04949 |