The $k$-dimensional Weisfeiler-Leman procedure ($k$-WL), which colors $k$-tuples of vertices in rounds based on the neighborhood structure in the graph, has proven to be immensely fruitful in the algorithmic study of Graph Isomorphism. More generally, it is of fundamental importance in understanding and exploiting symmetries in graphs in various settings. Two graphs are $k$-WL-equivalent if the $k$-dimensional Weisfeiler-Leman procedure produces the same final coloring on both graphs. 1-WL-equivalence is known as fractional isomorphism of graphs, and the $k$-WL-equivalence relation becomes finer as $k$ increases. We investigate to what extent standard graph parameters are preserved by $k$-WL-equivalence, focusing on fractional graph packing numbers. The integral packing numbers are typically NP-hard to compute, and we discuss applicability of $k$-WL-invariance for estimating the integrality gap of the LP relaxation provided by their fractional counterparts.

中文翻译：

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关于分数包装的 Weisfeiler-Leman 维数

$k$-Dimension Weisfeiler-Leman 过程 ($k$-WL)，它根据图中的邻域结构为 $k$-顶点元组在回合中着色，已被证明在图的算法研究中非常富有成效同构。更一般地说，它对于理解和利用各种设置中的图形对称性具有根本重要性。如果 $k$-Dimension Weisfeiler-Leman 过程在两个图上产生相同的最终着色，则两个图是 $k$-WL-等价的。1-WL-等价被称为图的分数同构，$k$-WL-等价关系随着$k$ 的增加而变得更精细。我们研究了 $k$-WL 等价在多大程度上保留了标准图参数，重点是分数图打包数。整数包装数通常是 NP 难计算的，