Correlation Bounds Against Monotone NC1

This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC(1) of polynomial-size O(log n)-depth monotone circuits. Our main theorem, proved using the pathset complexity framework introduced in [56], shows that the average-case k-CYCLE problem (on Erdos-Renyi random graphs with an appropriate edge density) is 1/2 + 1/ poly(n) hard for mNC(1). Combining this result with O'Donnell's hardness amplification theorem [43], we obtain an explicit monotone function of n variables (in the class mSAC(1)) which is 1/2+ n(-1/2+epsilon) hard for mNC(1) under the uniform distribution for any desired constant epsilon > 0. This bound is nearly best possible, since every monotone function has agreement 1/2 + Omega( log n/root n) with some function in mNC(1) [44]. Our correlation bounds against mNC(1) extend smoothly to non-monotone NC1 circuits with a bounded number of negation gates. Using Holley's monotone coupling theorem [30], we prove the following lemma: with respect to any product distribution, if a balanced monotone function f is 1/2 + delta hard for monotone circuits of a given size and depth, then f is 1/2 + (2(t+1) - 1)delta hard for (non-monotone) circuits of the same size and depth with at most t negation gates. We thus achieve a lower bound against NC1 circuits with (1/2 - epsilon) log n negation gates, improving the previous record of 1/6 log log n [7]. Our bound on negations is "half" optimal, since [log(n + 1) e negation gates are known to be fully powerful for NC1 [3, 21].