Efficient certificates for small-set vertex expansion

Develop polynomial-time verifiable certificates—such as low-degree sum-of-squares proofs—for small-set vertex expansion Ψ_δ(G) ≥ c (for some constant c > 1 and δ in (0, 1/2)), analogous to spectral certificates for edge expansion, so that algorithms can be applied to general graphs beyond those already admitting certified SSVE.

Background

The paper introduces small-set vertex expansion (SSVE) as a weaker notion than edge expansion and shows algorithms that work when graphs admit efficient certificates of SSVE. These certificates enable the use of sum-of-squares-based rounding techniques to find large independent sets.

Unlike edge expansion, where spectral methods provide natural certificates, the authors observe that analogous efficient certificates for vertex expansion are not currently available. They nonetheless demonstrate, as a proof of concept, that the noisy hypercube admits a non-trivially small degree SoS certificate of SSVE, highlighting the potential of such certificates if they can be developed more broadly.