关于稳定集问题精确子图层级不同版本的研究
On different Versions of the Exact Subgraph Hierarchy for the Stable Set Problem
AI总结：
本文研究了稳定集问题的精确子图层级（ESH），提出了一种基于更小半定规划的变体CESH，并引入了缩放型精确子图约束（SESC），证明了ESH的界始终优于或等于CESH的界。
摘要：
设 $G$ 为具有 $n$ 个顶点和 $m$ 条边的图。在求解 $G$ 的稳定数的多种层级方法中，精确子图层级（ESH）是其中之一。在第一层级，它将 Lovász theta 函数 $\vartheta(G)$ 计算为半定规划（SDP），该规划包含一个阶数为 $n+1$ 的矩阵变量以及 $n+m+1$ 个约束。在第 $k$ 层级，它将阶数为 $k$ 的子图的所有精确子图约束（ESC）添加到该 SDP 中。ESC 确保与子图对应的矩阵变量的子矩阵位于正确的多面体内。通过仅将部分 ESC 包含到 SDP 中，ESH 可以在计算上被有效利用。本文提出了一种 ESH 的变体，它通过一个阶数为 $n$ 且具有 $m+1$ 个约束的 SDP 来计算 $\vartheta(G)$。我们证明了将 ESC 包含到该 SDP 中是有意义的，并类似地提出了计算型 ESH（CESH）。在计算方面，由于 SDP 更小，CESH 似乎更具优势。然而，我们证明了基于 ESH 的界始终至少与基于 CESH 的界一样好。在计算实验中，有时 ESH 的界显著更好。我们还引入了缩放型 ESC（SESC），这是一种将精确约束包含到较小 SDP 中的更自然方式，并证明了包含一个 SESC 等价于为子图包含一个 ESC。
英文摘要：
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 Lovász 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
