We propose a simple and efficient local algorithm for graph isomorphism which succeeds for a large class of sparse graphs. This algorithm produces a low-depth canonical labeling, which is a labeling of the vertices of the graph that identifies its isomorphism class using vertices' local neighborhoods. Prior work by Czajka and Pandurangan showed that the degree profile of a vertex (i.e., the sorted list of the degrees of its neighbors) gives a canonical labeling with high probability when $n p_n = \omega( \log^{4}(n) / \log \log n )$ (and $p_{n} \leq 1/2$); subsequently, Mossel and Ross showed that the same holds when $n p_n = \omega( \log^{2}(n) )$. We first show that their analysis essentially cannot be improved: we prove that when $n p_n = o( \log^{2}(n) / (\log \log n)^{3} )$, with high probability there exist distinct vertices with isomorphic $2$-neighborhoods. Our first main result is a positive counterpart to this, showing that $3$-neighborhoods give a canonical labeling when $n p_n \geq (1+\delta) \log n$ (and $p_n \leq 1/2$); this improves a recent result of Ding, Ma, Wu, and Xu, completing the picture above the connectivity threshold. Our second main result is a smoothed analysis of graph isomorphism, showing that for a large class of deterministic graphs, a small random perturbation ensures that $3$-neighborhoods give a canonical labeling with high probability. While the worst-case complexity of graph isomorphism is still unknown, this shows that graph isomorphism has polynomial smoothed complexity.

翻译：我们提出一种简单高效的图同构局部算法，该算法对一大类稀疏图有效。该算法生成低深度规范标号，即通过顶点局部邻域识别图同构类的顶点标号。此前Czajka与Pandurangan的工作表明，当$n p_n = \omega( \log^{4}(n) / \log \log n )$（且$p_{n} \leq 1/2$）时，顶点的度数轮廓（即其邻居度数排序列表）以高概率给出规范标号；随后Mossel与Ross证明，当$n p_n = \omega( \log^{2}(n) )$时同样成立。我们首先证明其分析本质上不可改进：当$n p_n = o( \log^{2}(n) / (\log \log n)^{3} )$时，以高概率存在具有同构$2$-邻域的不同顶点。我们的第一个主要结果是该结论的正面补充，证明当$n p_n \geq (1+\delta) \log n$（且$p_n \leq 1/2$）时，$3$-邻域可给出规范标号；这改进了Ding、Ma、Wu与Xu的最新结果，完善了连通阈值之上的完整图景。第二个主要结果是图同构的平滑分析，表明对于一大类确定性图，小随机扰动可确保$3$-邻域以高概率给出规范标号。尽管图同构的最坏情况复杂度仍未知，但该结果表明图同构具有多项式平滑复杂度。