-

 

 

 




Optimization Online





 

Exactness of Parrilo's conic approximations for copositive matrices and associated low order bounds for the stability number of a graph

Monique Laurent(monique.laurent***at***cwi.nl)
Luis Felipe Vargas(luis.vargas***at***cwi.nl)

Abstract: De Klerk and Pasechnik (2002) introduced the bounds $\vartheta^{(r)}(G)$ ($r\in \mathbb{N}$) for the stability number $\alpha(G)$ of a graph $G$ and conjectured exactness at order $\alpha(G)-1$: $\vartheta^{(\alpha(G)-1)}(G)=\alpha(G)$. These bounds rely on the conic approximations $\mathcal{K}_n^{(r)}$ by Parrilo (2000) for the copositive cone $\text{COP}_n$. A difficulty in the convergence analysis of $\vartheta^{(r)}$ is the bad behaviour of the cones $\mathcal{K}_n^{(r)}$ under adding a zero row/column: when applied to a matrix not in $\mathcal{K}^{(0)}_n$ this gives a matrix not in any ${\mathcal{K}}^{(r)}_{n+1}$, thereby showing strict inclusion $\bigcup_{r\ge 0}{\mathcal{K}}^{(r)}_n\subset \text{COP}_n$ for $n\ge 6$. We investigate the graphs with $\vartheta^{(r)}(G)=\alpha(G)$ for $r=0,1$: we algorithmically reduce testing exactness of $\vartheta^{(0)}$ to acritical graphs, we characterize critical graphs with $\vartheta^{(0)}$ exact, and we exhibit graphs for which exactness of $\vartheta^{(1)}$ is not preserved under adding an isolated node. This disproves a conjecture by Gvozdenovi\'c and Laurent (2007) which, if true, would have implied the above conjecture by de Klerk and Pasechnik.

Keywords: stable set problem \and $\alpha$-critical graph \and sum-of-squares polynomial \and copositive matrix \and semidefinite programming \and Shor relaxation

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Category 2: Combinatorial Optimization (Approximation Algorithms )

Citation: arXiv:2109.12876

Download: [PDF]

Entry Submitted: 10/12/2021
Entry Accepted: 10/12/2021
Entry Last Modified: 10/12/2021

Modify/Update this entry


  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository

 

Submit
Update
Policies
Coordinator's Board
Classification Scheme
Credits
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society