We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lov\'{a}sz--Schrijver lift-and-project hierarchy. Our analysis of the relaxations of the stable set polytope leads to bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman, as well as the notion of deeply vertex-transitive graphs -- highly symmetric graphs that we show arise naturally from some association schemes. We also study relaxations of the hypergraph matching problem, and determine exactly or provide bounds on the lift-and-project ranks of these relaxations. Our proofs for these results also inspire the study of the general hypermatching pseudo-scheme, which is an association scheme except it is generally non-commutative. We then illustrate the usefulness of obtaining commutative subschemes from non-commutative pseudo-schemes via contraction in this context.

翻译：我们探索了结合方案与基于半正定规划（SDP）的Lovász-Schrijver提升-投影层次结构中组合优化问题凸松弛分析之间的一些联系。对稳定集多面体松弛的分析，得到了关于某些正则图的团数和稳定数上界，这些上界令人联想到Delsarte和Hoffman的经典界，以及深度顶点传递图的概念——我们证明这种高度对称的图自然源于某些结合方案。我们还研究了超图匹配问题的松弛，并精确确定了这些松弛的提升-投影秩或给出了其上界。这些结果的证明还启发了对一般超匹配伪方案的研究，该伪方案本质上是一种结合方案，但通常是非交换的。我们随后展示了在此背景下通过收缩从非交换伪方案中获取交换子方案的有用性。