Computational hardness of detecting graph lifts and certifying lift-monotone properties of random regular graphs

TL;DR

This paper conjectures that detecting random lifts of Ramanujan graphs is computationally hard and explores the implications of this conjecture for certifying properties of random regular graphs, supported by lower bounds against certain algorithms.

Contribution

It introduces a new conjecture on the hardness of detecting graph lifts and analyzes its implications for certifying properties of random regular graphs, supported by theoretical evidence.

Findings

Proves lower bounds against the local statistics hierarchy for hypothesis testing.

Shows no polynomial-time algorithm can certify tight bounds on maximum cut, independent set, or chromatic number under the conjecture.

Demonstrates spectral bounds are optimal among polynomial-time certificates for certain graph properties.

Abstract

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.