Correlation Bounds Against Monotone NC^1

This paper gives the first correlation bounds under product distributions (including the uni-form distribution) against the class mNC1 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 ap-propriate edge density) is 12 + 1 poly(n) hard for mNC 1. As a corollary, via O’Donnell’s hardness amplification theorem [43], we obtain an explicit monotone function of n variables (in the class mSAC1) which is 12 + n −1/2+ε hard for mNC1 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 mNC1 [44].