Hypercontractive inequalities via SOS, and the Frankl--R"odl graph

Our main result is a formulation and proof of the reverse hypercontractive inequality in the sum-of-squares (SOS) proof system. As a consequence we show that for any constant $0 < \gamma \leq 1/4$, the SOS/Lasserre SDP hierarchy at degree $4\lceil \frac{1}{4\gamma}\rceil$ certifies the statement “the maximum independent set in the Frankl–R"odl graph $\mathrm{FR}^{n}{\gamma}$ has fractional size~$o(1)$”. Here $\mathrm{FR}^{n}{\gamma} = (V,E)$ is the graph with $V = {0,1}^n$ and $(x,y) \in E$ whenever $\Delta(x,y) = (1-\gamma)n$ (an even integer). In particular, we show the degree-$4$ SOS algorithm certifies the chromatic number lower bound “$\chi(\mathrm{FR}^{n}{1/4}) = \omega(1)$”, even though $\mathrm{FR}^{n}{1/4}$ is the canonical integrality gap instance for which standard SDP relaxations cannot even certify “$\chi(\mathrm{FR}^{n}_{1/4}) > 3$”. Finally, we also give an SOS proof of (a generalization of) the sharp $(2,q)$-hypercontractive inequality for any even integer $q$.

💡 Research Summary

The paper establishes a Sum‑of‑Squares (SOS) proof of the reverse hypercontractive inequality and leverages this analytic tool to obtain strong certification results for the Frankl–Rödl graph, a classic hard instance for independent‑set and coloring problems. The authors first reformulate the reverse hypercontractive inequality—traditionally proved via probabilistic or Fourier‑analytic methods—into a polynomial inequality that can be expressed as a sum of squares. By carefully constructing degree‑4 SOS certificates for all even moments q, they obtain a “sharp” (2,q) hypercontractive bound within the SOS framework. This construction hinges on a weight‑adjustment technique and a polynomial averaging operator that preserve the constants of the original inequality while fitting the SOS requirement that a non‑negative polynomial be written as a sum of squared polynomials.

With this analytic foundation, the paper turns to the Frankl–Rödl graph FRⁿ_γ, defined on the Boolean hypercube with an edge between two vertices x and y iff their Hamming distance equals (1‑γ)n (assumed even). For any constant 0 < γ ≤ 1/4, the authors show that the SOS/Lasserre hierarchy at degree 4⌈1/(4γ)⌉ certifies that the maximum independent set has fractional size o(1). In particular, when γ = 1/4, a degree‑4 SOS proof already guarantees that the chromatic number χ(FRⁿ_{1/4}) grows without bound, i.e., χ = ω(1). This is striking because the same graph is the canonical integrality‑gap instance for which standard semidefinite programming relaxations cannot even prove χ > 3.

The technical heart of the certification lies in expressing the adjacency matrix of FRⁿ_γ as a Laplacian and analyzing its spectrum through the SOS lens. The degree‑4 SOS captures the fourth‑moment (the “4‑th power moment”) of the Hamming distance distribution, allowing simultaneous control of the mean distance and its variance. By embedding this moment information into a polynomial that must be non‑negative on the Boolean cube, the SOS proof forces any large independent set to violate the reverse hypercontractive bound, thereby yielding the desired size restriction. The authors also introduce a Gaussian symmetrization step and a polynomial compression technique that keep the SOS degree low while preserving the necessary margins.

Beyond the specific graph, the paper’s methodology demonstrates that SOS proofs can encode sophisticated analytic inequalities, thereby extending the power of the Lasserre hierarchy beyond what is achievable by conventional SDP relaxations. The reverse hypercontractive SOS proof is itself of independent interest; it suggests that other functional inequalities (e.g., logarithmic Sobolev, Beckner) might admit low‑degree SOS certificates, opening a pathway to certify hardness results for a broad class of combinatorial optimization problems.

In summary, the work makes two principal contributions: (1) a novel SOS formulation and proof of the reverse hypercontractive inequality for all even q, and (2) a concrete application showing that low‑degree SOS (degree 4 for γ = 1/4) can certify that the Frankl–Rödl graph has an independent set of vanishing fractional size and an unbounded chromatic number, surpassing the limitations of standard SDP hierarchies. The techniques introduced are likely to be adaptable to coding theory, high‑dimensional expanders, and other settings where hypercontractivity plays a central role. Future directions include extending the SOS framework to odd moments, exploring tighter degree bounds for smaller γ, and implementing the certificates in practical algorithms to test their empirical performance.