Finite convergence of sum-of-squares hierarchies for the stability number of a graph

Abstract

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

Figures (1)

• Figure 1: Graph $G$ (left), graph $H_1$ (middle), graph $H_2$ (right)

Theorems & Definitions (53)

• Conjecture 1: De Klerk and Pasechnik dKP2002
• Conjecture 2
• Example 1.1
• Theorem 2.1
• Theorem 2.2: see, e.g., Bertsekas/99
• Theorem 2.3: Nie Nie
• Lemma 3.1
• proof
• Lemma 3.2
• proof