Lower Bounding the AND-OR Tree via Symmetrization

07/15/2019

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by   William Kretschmer, et al.

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We prove a nearly tight lower bound on the approximate degree of the two-level AND-OR tree using symmetrization arguments. Specifically, we show that deg(AND_m ∘OR_n) = Ω(√(mn)). To our knowledge, this is the first proof of this fact that relies on symmetrization exclusively; most other proofs involve formulating approximate degree as a linear program and exhibiting an explicit dual witness. Our proof relies on a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson, Kothari, Kretschmer, and Thaler [AKKT19].