Spectrally Robust Graph Isomorphism

Abstract

We initiate the study of spectral generalizations of the graph isomorphism problem. b) 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)? c) The Spectrally Robust Graph Isomorphism (kappa-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. G preceq c H means that for all vectors x we have x^T L_G x <= c x^T L_H x, where L_G is the Laplacian G. We prove NP-hardness for SGD. We also present a kappa^3-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.