AI 中文总结
研究相关ER图对模型下的图匹配，提出基于局部树相关性测试、用基于秩算法匹配顶点对的图匹配算法，在特定假设下有$n^{2+o(1)}$时间复杂度，能高概率几乎精确恢复，是该$\lambda$范围首个近二次时间算法。
AI 中文摘要
本文研究相关的埃尔德什-雷尼（ER）图对模型下的图匹配。该模型先对一个$\mathrm{ER}(n,\frac{\lambda}{ns})$基础图进行采样，其边随后以概率$s$独立二次采样以生成两个相关的$\mathrm{ER}(n,\frac{\lambda}{n})$图。我们提出一种图匹配算法，在$\lambda=(\log n)^{\alpha+o(1)}$（对于某些$\alpha\in(0,1)$）和$s\in(\sqrt{C_{\mathrm{Otter}}},1]$的假设下，具有$n^{2+o(1)}$的时间复杂度且能以高概率实现几乎精确恢复，其中$C_{\mathrm{Otter}}\approx0.338$是奥特的树计数常数。该算法基于局部树相关性测试，使用基于秩的算法匹配顶点对，避免了计算难以获得的显式阈值。为证明几乎精确恢复结果，我们在平均度和树深度随$n$增长的发散度情况下对树相关性测试进行新分析，基于此结果建立基于阈值的图匹配算法的阈值存在性，最后结合基于秩的算法和基于阈值的算法性能以展示几乎精确恢复。
英文摘要
This paper studies graph matching under the correlated $\text{Erdős-Rényi}$ (ER) graph pair model. This model first samples an $\mathrm{ER}(n,\fracλ{ns})$ base graph, whose edges are then independently subsampled twice with probability $s$ to produce two correlated $\mathrm{ER}(n,\fracλ{n})$ graphs. We propose a graph matching algorithm that has $n^{2+o(1)}$ time complexity and achieves almost exact recovery with high probability under the assumptions $λ=(\log n)^{α+o(1)}$ for some $α\in(0,1)$ and $s\in(\sqrt{C_{\mathrm{Otter}}},1]$, where $C_{\mathrm{Otter}}\approx 0.338$ is Otter's tree-counting constant. This is the first algorithm with almost quadratic time complexity in this regime of $λ$, while the best known result in this regime is the chandelier-counting algorithm with time complexity $O(n^{c(s)})$, where $c(s)\rightarrow \infty$ as $s$ approaches $\sqrt{C_\mathrm{Otter}}$ from above. The proposed algorithm is based on local tree correlation tests. It uses a rank-based algorithm to match the vertex pairs instead of threshold-based rules in the literature. This avoids the need of computing an explicit threshold, which is computationally difficult to obtain. To prove the almost exact recovery result, we establish a new analysis of tree correlation tests in the diverging-degree regime, where both the mean degree and the tree depth grow with $n$. Based on this new result, we establish the existence of a threshold for a threshold-based graph matching algorithm via local tree correlation tests. Finally, we couple the performance of the rank-based algorithm with the threshold-based algorithm to show almost exact recovery.
CommentsAdded Acknowledgements; fixed a typo in the proof of Lemma 17
