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

Building on work of Cai, Fiirer, 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 nonisomor- phism requires degree Ω(n). In other words, we show an Ω (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 a-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 ε > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 - ε) isomorphic from pairs which are not even (1 - ε₀)-)- isomorphic for some universal constant ε₀)- Along the way we prove a robust asymmetry result for random graphs and hyper- graphs which may be of independent interest.