Depth Lower Bounds against Circuits with Sparse Orientation

We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function $f$ is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan circuit computing $f$. We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit $C$ computing the Clique function on an $n$ vertex graph. We prove that if $C$ is of depth $d$ and each gate computes a Boolean function with orientation of weight at most $w$ (in terms of the inputs to $C$), then $d \times w$ must be $\Omega(n)$. In particular, if the weights are $o(\frac{n}{\log^k n})$, then $C$ must be of depth $\omega(\log^k n)$. We prove a barrier for our general technique. However, using specific properties of the Clique function and the Karchmer-Wigderson framework (Karchmer and Wigderson, 1988), we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent. We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting), separates NP from NC. As our main tool, we generalize Karchmer-Wigderson gamefor monotone functions to work for non-monotone circuits parametrized by the weight/structure of the orientation. We also prove structural results about orientation and prove connections between number of negations and weight of orientations required to compute a function.