Notions of graph similarity provide alternative perspective on the graph isomorphism problem and vice-versa. In this paper, we consider measures of similarity arising from mismatch norms as studied in Gervens and Grohe: the edit distance $δ_{\mathcal{E}}$, and the metrics arising from $\ell_p$-operator norms, which we denote by $δ_p$ and $δ_{|p|}$. We address the following question: can these measures of similarity be used to design polynomial-time approximation algorithms for graph isomorphism? We show that computing an optimal value of $δ_{\mathcal{E}}$ is \NP-hard on pairs of graphs with the same number of edges. In addition, we show that computing optimal values of $δ_p$ and $δ_{|p|}$ is \NP-hard even on pairs of $1$-planar graphs with the same degree sequence and bounded degree. These two results improve on previous known ones, which did not examine the restricted case where the pairs of graphs are required to have the same number of edges. Finally, we study similarity problems on strongly regular graphs and prove some near optimal inequalities with interesting consequences on the computational complexity of graph and group isomorphism.

翻译：图相似性的概念为图同构问题提供了替代视角，反之亦然。本文研究了Gervens和Grohe提出的基于失配范数的相似性度量：编辑距离$δ_{\mathcal{E}}$，以及由$\ell_p$-算子范数导出的度量（分别记作$δ_p$和$δ_{|p|}$）。我们探讨以下问题：这些相似性度量能否用于设计图同构问题的多项式时间近似算法？研究表明，在边数相同的图对上计算$δ_{\mathcal{E}}$的最优值是\NP-困难的。此外，即使在具有相同度序列和有界度的$1$-平面图对上，计算$δ_p$和$δ_{|p|}$的最优值也是\NP-困难的。这两个结果改进了以往未考虑图对需具有相同边数这一受限情形的结论。最后，我们研究了强正则图上的相似性问题，并证明了若干近优不等式，这些不等式对图和群同构问题的计算复杂性具有重要启示。