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 that 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 this 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 of random 3- or 4-regular graphs, the maximum independent set of random 3- or 4-regular graphs, or the chromatic number of random 7-regular graphs. We show similar gaps asymptotically for large degree for the maximum independent set and for any degree for the minimum dominating set, finding that naive spectral and combinatorial bounds are optimal among all polynomial-time certificates. Likewise, for small-set vertex and edge expansion in the limit of very small sets, we show that the spectral bounds of Kahale (1995) are optimal among all polynomial-time certificates.

翻译：我们提出一个关于检测图随机提升计算难度的新猜想：声称不存在多项式时间算法能够区分大规模随机$d$-正则图与大规模拉马努金$d$-正则基图的随机提升（前提是该提升受到少量额外噪声干扰），同样适用于二分随机图与二分拉马努金图的提升。我们通过证明假设检验半定规划局部统计层次的下界为该猜想提供证据。随后，我们探讨该猜想对随机正则图多种函数认证界限计算难度的影响，拓展了Bandeira、Banks、Kunisky、Moore和Wein（2021）开创的研究方向。在该猜想成立的前提下，我们证明：不存在多项式时间算法能够认证随机3-或4-正则图的最大割的紧界、随机3-或4-正则图的最大独立集的紧界，或随机7-正则图的色数的紧界。对于最大独立集在大度数情形以及任意度数下的最小支配集，我们展示了相似的渐近差异，发现朴素谱界和组合界在所有多项式时间认证中是最优的。同样地，针对极小集合的顶点与边扩展性极限情形，我们证明Kahale（1995）的谱界在所有多项式时间认证中是最优的。