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 ×w must be Ω(n). In particular, if the weights are \(o(\frac{n}{\log^k n})\), then C must be of depth ω(log k n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in [4]) and the Karchmer-Wigderson framework [11], we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent.