The Lov\'asz theta function $\theta(G)$ provides a very good upper bound on the stability number of a graph $G$. It can be computed in polynomial time by solving a semidefinite program (SDP), which also turns out to be fairly tractable in practice. Consequently, $\theta(G)$ achieves a hard-to-beat trade-off between computational effort and strength of the bound. Indeed, several attempts to improve the theta bound are documented, mainly based on playing around the application of the $N_+(\cdot)$ lifting operator of Lov\'asz and Schrijver to the classical formulation of the maximum stable set problem. Experience shows that solving such SDP-s often struggles against practical intractability and requires highly specialized methods. We investigate the application of such an operator to two different linear formulations based on clique and nodal inequalities, respectively. Fewer inequalities describe these two and yet guarantee that the resulting SDP bound is at least as strong as $\theta(G)$. Our computational experience, including larger graphs than those previously documented, shows that upper bounds stronger than $\theta(G)$ can be accessed by a reasonable additional effort using the clique-based formulation on sparse graphs and the nodal-based one on dense graphs.

翻译：Lovász theta函数 $\theta(G)$ 为图 $G$ 的稳定数提供了一个极佳的上界。该函数可通过求解半定规划（SDP）在多项式时间内计算得出，且在实际中通常具有较好的可处理性。因此，$\theta(G)$ 在计算代价与界强度之间实现了难以超越的平衡。事实上，已有若干改进theta界的尝试被记录，主要基于对Lovász和Schrijver提出的 $N_+(\cdot)$ 提升算子应用于经典最大稳定集问题公式的探索。经验表明，求解此类SDP常面临实际可处理性挑战，并需要高度专业化的方法。我们研究了将该算子分别应用于基于团不等式和节点不等式的两种不同线性公式。这两种公式使用更少的不等式，同时保证所得SDP界至少与 $\theta(G)$ 一样强。我们的计算实验（包括比以往文献中更大的图）表明，在稀疏图上使用基于团的公式，在稠密图上使用基于节点的公式，通过合理的额外计算代价，可以获得比 $\theta(G)$ 更强的上界。