Computational Hardness of Detecting Graph Lifts and Certifying Lift-Monotone Properties of Random Regular Graphs

We introduce a new conjecture on the computational hardness of detecting random lifts of graphs: we claim that there is no polynomial-time algorithm that can distinguish between a large random $d$ -regular graph and a large random lift of a Ramanujan $d$ -regular base graph (provided the lift is corrupted by a small amount of extra noise), and likewise for bipartite random graphs and lifts of bipartite Ramanujan graphs. We give evidence for this conjecture by proving lower bounds against the local statistics hierarchy of hypothesis testing semidefinite programs. We then explore the consequences of the conjecture for the hardness of certifying bounds on numerous functions of random regular graphs, expanding on a direction initiated by Bandeira, Banks, Kunisky, Moore, and Wein (2021). Conditional on this conjecture, we show that no polynomial-time algorithm can certify tight bounds on the maximum cut or maximum independent set of random 3- or 4-regular graphs, or on the chromatic number of random 7-regular graphs. Asymptotically for large degree for the maximum independent set and for any degree for the minimum dominating set, we show similar gaps, finding that naive spectral and combinatorial bounds are optimal among efficiently computable ones. Likewise, for small set vertex and edge expansion in the limit of very small sets, we show that the spectral bounds due to Kahale (1995) are optimal efficient certificates.