Dual lower bounds for approximate degree and markov-bernstein inequalities

  • Authors:

  • Mark Bun;Justin Thaler

  • Affiliations:

  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA;School of Engineering and Applied Sciences, Harvard University, Cambridge, MA

  • Venue:
  • ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

The ε-approximate degree of a Boolean function f: {−1, 1}n→{−1, 1} is the minimum degree of a real polynomial that approximates f to within ε in the ℓ∞ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the ε-approximate degree of the two-level AND-OR tree for any constant ε0. We show that this quantity is $\Theta(\sqrt{n})$, closing a line of incrementally larger lower bounds [3,11,21,30,32]. The same lower bound was recently obtained independently by Sherstov using related techniques [25]. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek [34]. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.