Let $G$ be a graph with $n$ vertices and $m$ edges. One of several hierarchies towards the stability number of $G$ is the exact subgraph hierarchy (ESH). On the first level it computes the Lov\'{a}sz theta function $\vartheta(G)$ as semidefinite program (SDP) with a matrix variable of order $n+1$ and $n+m+1$ constraints. On the $k$-th level it adds all exact subgraph constraints (ESC) for subgraphs of order $k$ to the SDP. An ESC ensures that the submatrix of the matrix variable corresponding to the subgraph is in the correct polytope. By including only some ESCs into the SDP the ESH can be exploited computationally. In this paper we introduce a variant of the ESH that computes $\vartheta(G)$ through an SDP with a matrix variable of order $n$ and $m+1$ constraints. We show that it makes sense to include the ESCs into this SDP and introduce the compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems favorable as the SDP is smaller. However, we prove that the bounds based on the ESH are always at least as good as those of the CESH. In computational experiments sometimes they are significantly better. We also introduce scaled ESCs (SESCs), which are a more natural way to include exactness constraints into the smaller SDP and we prove that including an SESC is equivalent to including an ESC for every subgraph.

翻译：设$G$是一个具有$n$个顶点和$m$条边的图。针对图$G$的稳定数，精确子图层次结构（ESH）是若干层次结构之一。在第一层次，它通过半定规划（SDP）计算Lovász theta函数$\vartheta(G)$，该SDP的矩阵变量阶数为$n+1$，并包含$n+m+1$个约束。在第$k$层次，它在SDP中增加所有关于$k$阶子图的精确子图约束（ESC）。ESC确保矩阵变量中对应于该子图的子矩阵位于正确的多面体内。通过仅将部分ESC纳入SDP，可以在计算上利用ESH。本文提出一种ESH的变体，它通过一个矩阵变量阶数为$n$、包含$m+1$个约束的SDP来计算$\vartheta(G)$。我们证明，将该SDP纳入ESC是合理的，并类比ESH引入压缩精确子图层次结构（CESH）。从计算角度看，CESH似乎更具优势，因为其SDP规模更小。然而，我们证明基于ESH的界始终不低于CESH的界。在计算实验中，前者有时显著优于后者。我们还引入了缩放精确子图约束（SESC），这是一种将精确性约束纳入较小SDP的更自然方式，并证明纳入一个SESC等价于为每个子图纳入一个ESC。