Correlation Bounds Against Monotone NC

This paper gives the first correlation bounds under product distributions (including the uniform distribution) against the class mNC of poly(n)-size O(log n)-depth monotone circuits. Our main theorem, proved using the pathset complexity framework recently introduced in [56], shows that the average-case k-CYCLE problem (on Erdős-Rényi random graphs with an appropriate edge density) is 12 + 1 poly(n) hard for mNC . As a corollary, via O’Donnell’s hardness amplification theorem [43], we obtain an explicit monotone function of n variables (in the class mSAC) which is 12 + n −1/2+ε hard for mNC under the uniform distribution, for any desired ε > 0. (This bound is nearly tight, since every monotone function has correlation Ω( logn √ n ) with a function in mNC [44].) Unlike previous lower bounds for monotone circuits (i.e. under non-product distributions), these correlation bounds extend smoothly to negation-limited circuits. By a simple argument using Holley’s monotone coupling theorem [30], we show the following lemma: under any product distribution, if a balanced monotone function f is 1 2 +δ hard for monotone circuits of a given size and depth, then f is 12 + (2 t+1 − 1)δ hard, up to the same size and depth, for (non-monotone) boolean circuits with t negation gates. Our correlation bounds against mNC thus extend to NC circuits with ( 12 − ε) log n negations. (This improves a previous 1 6 log log n lower bound [7] on the negation-limited complexity of an explicit monotone function; by [21], NC circuits with dlog(n+ 1)e negations are equivalent to full NC.)