Spectrally Robust Graph Isomorphism

TL;DR

This paper introduces spectral generalizations of graph isomorphism, focusing on the Spectral Graph Dominance and Spectrally Robust Graph Isomorphism problems, with complexity results and approximation algorithms for specific graph classes.

Contribution

It formalizes spectral graph isomorphism variants, proves NP-hardness for Spectral Graph Dominance, and provides a polynomial-time approximation algorithm for bounded-degree trees in the SRGI problem.

Findings

NP-hardness of Spectral Graph Dominance

Approximation algorithm for SRGI on bounded-degree trees

Polynomial-time algorithm when approximation factor is constant

Abstract

We initiate the study of spectral generalizations of the graph isomorphism problem. (a)The Spectral Graph Dominance (SGD) problem: On input of two graphs $G$ and $H$ does there exist a permutation $\pi$ such that $G\preceq \pi(H)$? (b) The Spectrally Robust Graph Isomorphism (SRGI) problem: On input of two graphs $G$ and $H$, find the smallest number $\kappa$ over all permutations $\pi$ such that $ \pi(H) \preceq G\preceq \kappa c \pi(H)$ for some $c$. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology. Here $G\preceq c H$ means that for all vectors $x$ we have $x^T L_G x \leq c x^T L_H x$, where $L_G$ is the Laplacian $G$. We prove NP-hardness for SGD. We also present a $\kappa$-approximation algorithm for SRGI for the case when both $G$ and $H$ are bounded-degree trees. The algorithm runs in polynomial time when $\kappa$ is a constant.