Given two graphs which are almost isomorphic, is it possible to find a bijection which preserves most of the edges between the two? This is the algorithmic task of Robust Graph Isomorphism, which is a natural approximation variation of the Graph Isomorphism problem. In this talk, we show that no polynomial-time algorithm solves this problem, conditioned on Feige's Random 3XOR Hypothesis. In addition, we show that the Lasserre/SOS SDP hierarchy, the most powerful SDP hierarchy known, fails quite spectacularly on this problem: it needs a linear number of rounds to distinguish two isomorphic graphs from two far-from-isomorphic graphs. Along the way, we venture into the theory of random graphs by showing that a random graph is robustly asymmetric whp, meaning that any permutation which is close to an automorphism is itself close to the identity permutation.
Joint work with Ryan O'Donnell, John Wright, and Chenggang Wu.