Correlation Bounds Against Monotone NC^1

This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC1 of polynomial-size O(logn)-depth monotone circuits. Our main theorem, proved using the pathset complexity framework introduced in [56], shows that the average-case k-CYCLE problem (on Erdős-Rényi random graphs with an appropriate edge density) is 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 mSAC1) which is 2 +n −1/2+ε hard for mNC1 under the uniform distribution for any desired constant ε > 0. This bound is nearly best possible, since every monotone function has agreement 2 + Ω( logn √ n ) with some function in mNC1 [44]. Our correlation bounds against mNC1 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 + δ hard for monotone circuits of a given size and depth, then f is 1 2 + (2 t+1 − 1)δ 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 − ε) logn negation gates, improving the previous record of 6 log logn [7]. Our bound on negations is “half” optimal, since dlog(n + 1)e negation gates are known to be fully powerful for NC1 [3, 21]. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes