稀疏图中独立集的Lovász θ函数
On the Lovász Theta function for Independent Sets in Sparse Graphs
摘要：
本文研究最大度为~ $d$的图的最大独立集问题。我们证明了基于$\vartheta$函数的SDP的完整性间隙是$\widetilde{O}(d/\log^{3/2} d)$ .这改进了$\widetilde{O}(d/\log d)$的先前最佳结果，并且几乎匹配最近针对更强的SDP所示的$\widetilde{O}(d/\log^2 d)$的完整性间隙，即使用$SA^+$半定层级的聚$(\log(d))$水平获得的那些。改进来自于改进的Ramsey理论界的独立数的$K_r$ -自由图的大值$r$。我们还展示了如何通过Johansson的着色算法获得上述基于$SA^+$的完整性间隙结果的算法版本。得到的$\widetilde{O}(d/\log^2 d)$的近似保证匹配最佳的基于唯一游戏的硬度结果，直到低阶多$(\log\log d)$因子。
英文摘要：
We consider the maximum independent set problem on graphs with maximum degree~$d$. We show that the integrality gap of the Lovász $\vartheta$-function based SDP is $\widetilde{O}(d/\log^{3/2} d)$. This improves on the previous best result of $\widetilde{O}(d/\log d)$, and almost matches the integrality gap of $\widetilde{O}(d/\log^2 d)$ recently shown for stronger SDPs, namely those obtained using poly-$(\log(d))$ levels of the $SA^+$ semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of $K_r$-free graphs for large values of $r$. We also show how to obtain an algorithmic version of the above-mentioned $SA^+$-based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of $\widetilde{O}(d/\log^2 d)$ matches the best unique-games-based hardness result up to lower-order poly-$(\log\log d)$ factors.
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