Sum-of-squares lower bounds for constant-degree hierarchies on random regular graphs

Prove sum-of-squares lower bounds for arbitrary constant degrees of the hierarchy for any of the graph function certification tasks considered in the paper (e.g., maximum cut, independence number, chromatic number, domination number, expansion) on random d-regular graphs with constant d as n→∞.

Background

Beyond the local statistics (LoSt) hierarchy, the paper discusses the longstanding difficulty of proving lower bounds against higher-degree sum-of-squares (SoS) relaxations, especially on sparse random regular graphs. Establishing such lower bounds would provide unconditional evidence of computational hardness for certifying tight bounds on numerous graph properties without relying on the lift-detection conjecture.

The authors explicitly identify this as an open problem and note its relevance across the various quantities they analyze (max-cut, independent set, chromatic number, dominating set, vertex/edge expansion) for constant-degree regular graphs.