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Finite convergence of sum-of-squares hierarchies for the stability number of a graph
We investigate a hierarchy of semidefinite bounds for the stability number of a graph , based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM J. Optim., 12 (2002), pp. 875--892], who conjectured convergence to in steps. Even the weaker conjecture claiming finite convergence is still open. We establish links between this hierarchy and sum-of-squares hierarchies based on the Motzkin--Straus formulation of , which we use to show finite convergence when is acritical, i.e., when for all edges of . This relies, in particular, on understanding the structure of the minimizers of the Motzkin--Straus formulation and showing that their number is finite precisely when is acritical. Moreover we show that these results hold in the general setting of the weighted stable set problem for graphs equipped with positive node weights. In addition, as a byproduct we show that deciding whether a standard quadratic program has finitely many minimizers does not admit a polynomial-time algorithm unless P=NP