Hypercontractive inequalities via SOS, and the Frankl-Rödl graph

Manuel Kauers; Ryan O'Donnell; Li Yang Tan; Yuan Zhou

Hypercontractive inequalities via SOS, and the Frankl-Rödl graph

Manuel Kauers, Ryan O'Donnell, Li Yang Tan, Yuan Zhou

Research output: Contribution to journal › Article › peer-review

Abstract

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 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree 4[1/4gamma;]certifies the statement "the maximum independent set in the Frankl-Rödl graph FR_γⁿ has fractional size o(1)''.Here FR_γⁿ = (V, E) is the graph with V = (0,1)ⁿ and (x, y) ∈ E wheneverΔ(x, y) = (1-γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound "Χ(FRⁿ _1/4) = ω(1)", relaxations cannot even certify "Χ(FR_1/4ⁿ > 3ⁿ. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

Original language	English (US)
Pages (from-to)	1-21
Number of pages	21
Journal	Discrete Analysis
Volume	4
Issue number	2016
State	Published - 2016
Externally published	Yes

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics

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AU - Kauers, Manuel

AU - O'Donnell, Ryan

AU - Tan, Li Yang

AU - Zhou, Yuan

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Y1 - 2016

N2 - 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 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree 4[1/4gamma;]certifies the statement "the maximum independent set in the Frankl-Rödl graph FRγn has fractional size o(1)''.Here FRγn = (V, E) is the graph with V = (0,1)n and (x, y) ∈ E wheneverΔ(x, y) = (1-γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound "Χ(FRn 1/4) = ω(1)", relaxations cannot even certify "Χ(FR1/4n > 3n. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

AB - 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 < γ ≤ 1/4, the SOS/Lasserre SDP hierarchy at degree 4[1/4gamma;]certifies the statement "the maximum independent set in the Frankl-Rödl graph FRγn has fractional size o(1)''.Here FRγn = (V, E) is the graph with V = (0,1)n and (x, y) ∈ E wheneverΔ(x, y) = (1-γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound "Χ(FRn 1/4) = ω(1)", relaxations cannot even certify "Χ(FR1/4n > 3n. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)-hypercontractive inequality for any even integer q.

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Hypercontractive inequalities via SOS, and the Frankl-Rödl graph

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