Corpus ID: 237940398

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

@inproceedings{Laurent2021ExactnessOP,
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},
year={2021}
}
• Published 27 September 2021
• Mathematics
De Klerk and Pasechnik (2002) introduced the bounds θ(G) (r ∈ N) for the stability number α(G) of a graph G and conjectured exactness at order α(G) − 1: θ(G) = α(G). These bounds rely on the conic approximations K (r) n by Parrilo (2000) for the copositive cone COPn. A difficulty in the convergence analysis of θ is the bad behaviour of the cones K (r) n under adding a zero row/column: when applied to a matrix not in K (0) n this gives a matrix not in any K (r) n+1, thereby showing strict…

References

SHOWING 1-10 OF 31 REFERENCES
Semidefinite Bounds for the Stability Number of a Graph via Sums of Squares of Polynomials
• Mathematics, Computer Science
IPCO
• 2005
The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovasz and Schrijver, and the de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α( G) steps is proved.
Approximation of the Stability Number of a Graph via Copositive Programming
• Mathematics, Computer Science
SIAM J. Optim.
• 2002
This paper shows how the stability number can be computed as the solution of a conic linear program (LP) over the cone of copositive matrices of a graph by solving semidefinite programs (SDPs) of increasing size (lift-and-project method).
Computing the Stability Number of a Graph Via Linear and Semidefinite Programming
• Computer Science, Mathematics
SIAM J. Optim.
• 2007
This work is based on and refines de Klerk and Pasechnik’s approach to approximating the stability number via copositive programming and provides a closed-form expression for the values computed by the linear programming approximations.
Continuous Characterizations of the Maximum Clique Problem
• Mathematics, Computer Science
Math. Oper. Res.
• 1997
The Motzkin-Strauss QP is characterized, revealing interesting underlying discrete structures, and its properties are investigated, which are polynomial time verifiable.
Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs
• Mathematics, Computer Science
Math. Program.
• 2020
This work studies a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs, and provides a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact, the first result establishing the exactness of the semideFinite relaxation for random general QCZPs.
Kneser's Conjecture, Chromatic Number, and Homotopy
• L. Lovász
• Computer Science, Mathematics
J. Comb. Theory, Ser. A
• 1978
Abstract If the simplicial complex formed by the neighborhoods of points of a graph is (k − 2)-connected then the graph is not k-colorable. As a corollary Kneser's conjecture is proved, asserting
An exact duality theory for semidefinite programming and its complexity implications
• M. Ramana
• Mathematics, Computer Science
Math. Program.
• 1997
In this paper, an exact dual is derived for Semidefinite Programming (SDP), for which strong duality properties hold without any regularity assumptions, and the dual is then applied to derive certain complexity results for SDP.
Maxima for Graphs and a New Proof of a Theorem of Turán
• Mathematics
• 1965
Maximum of a square-free quadratic form on a simplex. The following question was suggested by a problem of J. E. MacDonald Jr. (1): Given a graph G with vertices 1, 2, . . . , n. Let S be the simplex
Scaling relationship between the copositive cone and Parrilo’s first level approximation
• Mathematics, Computer Science
Optim. Lett.
• 2013
The relation between the cone of n × n copositive matrices and the approximating cone introduced by Parrilo is investigated and it is shown that for n ≥ 5 they are not equal.
On the Shannon capacity of a graph
• L. Lovász
• Mathematics, Computer Science
IEEE Trans. Inf. Theory
• 1979
It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.