Lower Bounding the AND-OR Tree via Symmetrization

William Kretschmer

doi:10.1145/3434385

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arXiv · 1907.06731

Lower Bounding the AND-OR Tree via Symmetrization

William Kretschmer

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Abstract

We prove a simple, nearly tight lower bound on the approximate degree of the two-level $\mathsf{AND}$-$\mathsf{OR}$ tree using symmetrization arguments. Specifically, we show that $\widetilde{\mathrm{deg}}(\mathsf{AND}_m \circ \mathsf{OR}_n) = \widetilde{\Omega}(\sqrt{mn})$. We prove this lower bound via reduction to the $\mathsf{OR}$ function through a series of symmetrization steps, in contrast to most other proofs that involve formulating approximate degree as a linear program [BT13, She13, BDBGK18]. Our proof also demonstrates the power of a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson, Kothari, Kretschmer, and Thaler [AKKT19].

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William Kretschmer. 2019-07-15. Lower Bounding the AND-OR Tree via Symmetrization. https://doi.org/10.1145/3434385

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