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

Monique Laurent () and Luis Felipe Vargas ()
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Monique Laurent: Centrum Wiskunde & Informatica, 1098 SJ Amsterdam, Netherlands; Tilburg University, 5037 AB Tilburg, Netherlands
Luis Felipe Vargas: Centrum Wiskunde & Informatica, 1098 SJ Amsterdam, Netherlands

Mathematics of Operations Research, 2023, vol. 48, issue 2, 1017-1043

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.

Keywords: Primary: 05Cxx; 90C22; 90C23; 90C26; 90C27; 11E25; stable set problem; α-critical graph; sum-of-squares polynomial; copositive matrix; semidefinite programming; Shor relaxation; polynomial optimization (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:48:y:2023:i:2:p:1017-1043

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