Improved Separations between Nondeterministic and Randomized Multiparty Communication
ACM Transactions on Computation Theory (TOCT)
Optimal bounds for sign-representing the intersection of two halfspaces by polynomials
Proceedings of the forty-second ACM symposium on Theory of computing
SIAM Journal on Computing
On quantum-classical equivalence for composed communication problems
Quantum Information & Computation
The multiparty communication complexity of set disjointness
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Making polynomials robust to noise
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
SIAM Journal on Computing
Communication lower bounds using directional derivatives
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Dual lower bounds for approximate degree and markov-bernstein inequalities
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We construct a function in ${AC}^0$ that cannot be computed by a depth-2 majority circuit of size less than $\exp(\Theta(n^{1/5}))$. This solves an open problem due to Krause and Pudlák [Theoret. Comput. Sci., 174 (1997), pp. 137-156] and matches Allender's classic result [A note on the power of threshold circuits, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Research Triangle Park, NC, 1989, pp. 580-584] that ${AC}^0$ can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This technique, the Degree/Discrepancy Theorem, is of independent interest. It translates lower bounds on the threshold degree of any Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply lower bounds on communication and circuit size. In particular, we exhibit the first known function in ${AC}^0$ with exponentially small discrepancy, $\exp(-\Omega(n^{1/5}))$, thereby establishing the separations $\Sigma_2^{cc}\not\subseteq{PP}^{cc}$ and $\Pi_2^{cc}\not\subseteq{PP}^{cc}$ in communication complexity.