Year: 2026
Type: article
Abstract: 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.
DOI: 10.46298/dmtcs.12508
Authors: He Sun, Danny Vagnozzi
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4386555392