FINITE CONVERGENCE OF SUM-OF-SQUARES HIERARCHIES FOR THE STABILITY NUMBER OF A GRAPH

We investigate a hierarchy of semidefinite bounds nu((r))(G) for the stability number alpha(G) of a graph G, based on its copositive programming formulation and introduced by de Klerk and Pasechnik [SIAM T. Optim., 12 (2002), pp. 875-892], who conjectured convergence to alpha(G) in r = alpha(G)-1 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 alpha(G), which we use to show finite convergence when G is acritical, i.e., when alpha(G \ e) = alpha(G) for all edges e of G. 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 G 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.