Hardness of Robust Graph Isomorphism, Lasserre Gaps, and Asymmetry of Random Graphs
Ryan W. O’Donnell, John C. Wright, Chenggang Wu, Yuan Zhou · 2013
Building on work of Cai, Fürer, 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 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 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 ∊ > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 — ∊)-isomorphic from pairs which are not even (1 – ∊0)-isomorphic for some universal constant ∊0. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest.