Quantified Derandomization of Linear Threshold Circuits

One of the prominent current challenges in complexity theory is the attempt to prove lower bounds for T C, the class of constant-depth, polynomial-size circuits with majority gates. Relying on the results of Williams (2013), an appealing approach to prove such lower bounds is to construct a non-trivial derandomization algorithm for T C. In this work we take a first step towards the latter goal, by proving the first positive results regarding the derandomization of T C circuits of depth d > 2. Our first main result is a quantified derandomization algorithm for T C circuits with a super-linear number of wires. Specifically, we construct an algorithm that gets as input a T C circuit C over n input bits with depth d and n1+exp(−d) wires, runs in almostpolynomial-time, and distinguishes between the case that C rejects at most 2n 1−1/5d inputs and the case that C accepts at most 2n 1−1/5d inputs. In fact, our algorithm works even when the circuit C is a linear threshold circuit, rather than just a T C circuit (i.e., C is a circuit with linear threshold gates, which are stronger than majority gates). Our second main result is that even a modest improvement of our quantified derandomization algorithm would yield a non-trivial algorithm for standard derandomization of all of T C, and would consequently imply that NEXP 6⊆ T C. Specifically, if there exists a quantified derandomization algorithm that gets as input a T C circuit with depth d and n1+O(1/d) wires (rather than n1+exp(−d) wires), runs in time at most 2n exp(−d) , and distinguishes between the case that C rejects at most 2n 1−1/5d inputs and the case that C accepts at most 2n 1−1/5d inputs, then there exists an algorithm with running time 2n 1−Ω(1) for standard derandomization of T C. ∗Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Email: [email protected]