Exactness of Parrilo's conic approximations for copositive matrices >> and associated low order bounds for the stability number of a graph
Laurent,Monique Vargas,Luis
Laurent,Monique
Vargas,Luis
Abstract
De Klerk and Pasechnik introduced in 2002 semidefinite bounds ϑ (r)(G)(r ≥ 0) for the stability number α(G) of a graph G and conjectured their exactness at order r = α(G) − 1. These bounds rely on the conic approximations K ( n r) introduced by Parrilo in 2000 for the copositive cone COP n. 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 in K ( n r) this gives a matrix that does not lie in any of Parrilo’s cones, thereby showing that their union is a strict subset of the copositive cone for any n ≥ 6. We investigate the graphs for which the bound of order r ≤ 1 is exact: we algorithmically reduce testing exactness of ϑ (0) to acritical graphs, we characterize the critical graphs with ϑ (0) exact, and we exhibit graphs for which exactness of ϑ (1) is not preserved under adding an isolated node. This disproves a conjecture posed by Gvozdenović and Laurent in 2007, which, if true, would have implied the above conjecture by de Klerk and Pasechnik.
Description
Publisher Copyright: © 2022 INFORMS.
Date
2023-05
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Volume Title
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Research Projects
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Keywords
Shor relaxation, a-critical graph, copositive matrix, polynomial optimization, semidefinite programming, stable set problem, sum-of-squares polynomial
Citation
Laurent, M & Vargas, L 2023, 'Exactness of Parrilo's conic approximations for copositive matrices > > and associated low order bounds for the stability number of a graph', Mathematics of Operations Research, vol. 48, no. 2, pp. 1017-1043. https://doi.org/10.1287/moor.2022.1290
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info:eu-repo/semantics/openAccess