Three Hardness Results for Graph Similarity Problems

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 $\delta_{\mathcal{E}}$, and the metrics arising from $\ell_p$-operator norms, which we denote by $\delta_p$ and $\delta_{|p|}$. 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 $\delta_{\mathcal{E}}$ is \NP-hard on pairs of graphs with the same number of edges. In addition, we show that computing optimal values of $\delta_p$ and $\delta_{|p|}$ 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.