关于结合方案与多面体及半正定提升投影松弛分析的联系
On Connections Between Association Schemes and Analyses of Polyhedral and Positive Semidefinite Lift-and-Project Relaxations
AI总结：
该研究探索结合方案与Lovász-Schrijver层次中基于SDP的凸松弛分析的联系，给出正则图团数和稳定数的界，提出深度顶点传递图概念，研究超图匹配松弛的秩并说明收缩的作用。
摘要：
我们探索了结合方案与Lovász-Schrijver提升投影层次中基于半定规划（SDP）的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出了某些正则图的团数和稳定数的界，这些界让人联想到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产生于某些结合方案。我们还研究了超图匹配问题的松弛，精确确定或给出了这些松弛的提升投影秩的界。这些结果的证明也启发了基于超匹配的齐次相干构型的研究，该构型除通常非交换外，其余均为结合方案。然后我们通过收缩说明了在此背景下从非交换齐次相干构型中获得交换子方案的有用性。
英文摘要：
We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lovász--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 a homogeneous coherent configuration based on hypermatchings, which is an association scheme except it is generally non-commutative. We then illustrate the usefulness of obtaining commutative subschemes from non-commutative homogeneous coherent configurations via contraction in this context.
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