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，且在实践中也表现出良好的可处理性。因此，$\theta(G)$ 在计算成本与边界强度之间实现了难以超越的权衡。事实上，已有若干改进theta边界的尝试被记录在案，主要基于将Lovász和Schrijver的 $N_+(\cdot)$ 提升算子应用于经典的最大稳定集问题表述。经验表明，求解此类SDP常常面临实际可处理性的挑战，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规划表述。这两种表述使用的约束更少，但仍能保证所得的SDP边界至少与 $\theta(G)$ 一样强。我们的计算实验（包括比先前文献中更大的图）表明，通过在稀疏图上使用基于团的表述、在稠密图上使用基于节点的表述，可以以合理的额外计算成本获得比 $\theta(G)$ 更强的上界。