We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lov{\'a}sz--Schrijver SDP operator $\LS_+$, with a particular focus on a search for relatively small graphs with high $\LS_+$-rank (the least number of iterations of the $\LS_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\LS_+$-rank is asymptotically a linear function of its number of vertices, which is the best possible up to improvements in the constant factor (previous best result in this direction, from 1999, yielded graphs whose $\LS_+$-rank only grew with the square root of the number of vertices).

翻译：我们研究图稳定集多胞体关于Lovász–Schrijver SDP算子$\LS_+$的提升投影秩，重点关注寻找具有高$\LS_+$-秩（从分数稳定集多胞体出发，经$\LS_+$算子迭代计算稳定集多胞体所需的最少迭代次数）的相对小规模图。我们构造了图族，其$\LS_+$-秩渐近地随顶点数线性增长，这是除常数因子优化外可能达到的最佳结果（此前该方向的最佳结果为1999年得到的图，其$\LS_+$-秩仅随顶点数的平方根增长）。