Spectrally Robust Graph Isomorphism

• Published in: arXiv.org
• Main Authors: Kolla, Alexandra, Koutis, Ioannis, Madan, Vivek, Sinop, Ali Kemal
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 01.05.2018
• Subjects: Algorithms; Computer Science - Data Structures and Algorithms; Graphs; Isomorphism; Permutations; Polynomials; Spectra
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1805.00181

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.