Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs
Building on work of Cai, Fürer, and Immerman \cite{CFI92}, 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 $Ω(n)$-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 $α$-isomorphic if there is a bijection between their vertices which preserves at least $αm$ edges.) Our second result is that under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust} Graph Isomorphism problem 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 $(1-ε_0)$-isomorphic for some universal constant $ε_0$. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest.