Hardness of robust graph isomorphism, lasserre gaps, and asymmetry of random graphs

Building on work of Cai, Furer, and Immerman [18], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n-vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomorphism requires degree Ω(n). In other words, we show an O(n)-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1--10-14)-isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are α-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [23] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism is hard. I.e. for every e > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 -- e)-isomorphic from pairs which are not even (1 -- e0)-isomorphic for some universal constant e0. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest.