# Exactness of Parrilo’s Conic Approximations for Copositive Matrices and Associated Low Order Bounds for the Stability Number of a Graph

@article{Laurent2022ExactnessOP, title={Exactness of Parrilo’s Conic Approximations for Copositive Matrices and Associated Low Order Bounds for the Stability Number of a Graph}, author={Monique Laurent and Luis Felipe Vargas}, journal={Mathematics of Operations Research}, year={2022} }

De Klerk and Pasechnik introduced in 2002 semidefinite bounds [Formula: see text] for the stability number [Formula: see text] of a graph G and conjectured their exactness at order [Formula: see text]. These bounds rely on the conic approximations [Formula: see text] introduced by Parrilo in 2000 for the copositive cone [Formula: see text]. A difficulty in the convergence analysis of the bounds is the bad behavior of Parrilo’s cones under adding a zero row/column: when applied to a matrix not…

## 5 Citations

### Finite Convergence of Sum-of-Squares Hierarchies for the Stability Number of a Graph

- MathematicsSIAM J. Optim.
- 2022

We investigate a hierarchy of semidefinite bounds θ(G) for the stability number α(G) of a graph G, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J.…

### On the Exactness of Sum-of-Squares Approximations for the Cone of $5\times 5$ Copositive Matrices

- Mathematics
- 2022

We investigate the hierarchy of conic inner approximations K ( r ) n ( r ∈ N ) for the copositive cone COP n , introduced by Parrilo ( Structured Semideﬁnite Programs and Semialgebraic Geometry…

### On the exactness of sum-of-squares approximations for the cone of 5 × 5 copositive matrices

- MathematicsLinear Algebra and its Applications
- 2022

### On the structure of the $6 \times 6$ copositive cone

- Mathematics
- 2022

In this work we complement the description of the extreme rays of the 6 × 6 copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone…

### A Sum of Squares Characterization of Perfect Graphs

- Mathematics
- 2021

We present an algebraic characterization of perfect graphs, i.e., graphs for which the clique number and the chromatic number coincide for every induced subgraph. We show that a graph is perfect if…

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### Finite Convergence of Sum-of-Squares Hierarchies for the Stability Number of a Graph

- MathematicsSIAM J. Optim.
- 2022

We investigate a hierarchy of semidefinite bounds θ(G) for the stability number α(G) of a graph G, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J.…

### On the exactness of sum-of-squares approximations for the cone of 5 × 5 copositive matrices

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- 2022

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