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 certifies the statement “the maximum independent set in the Frankl–Rödl graph has fractional size o(1)”. Here is the graph with V = {0,1}n and (x,y) ∊ E whenever Δ(x, y) = (1 – γ)n (an even integer). In particular, we show the degree-4 SOS algorithm certifies the chromatic number lower bound “ ”, even though is the canonical integrality gap instance for which standard SDP relaxations cannot even certify “ ”. 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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