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 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.
Original language | English (US) |
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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