| Title | : | Depth Lower Bounds against Circuits with Sparse Orientation |
| Speaker | : | Sajin Koroth (IITM) |
| Details | : | Tue, 7 Jul, 2015 2:15 PM @ BSB 361 |
| Abstract: | : | Proving size/depth lower bounds against resource bounded Boolean circuits is the holy grail of circuit complexity theory. Strong lower bounds are known against circuits which are monotone (i.e., circuits without any negation gates). But very little is known about general (i.e., non-monotone) circuits.
Our work addresses 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 (circuits where negations appear only at the leaves) 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 x 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, 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. We demonstrate that this approach reaches out to the limits in terms of depth lower bounds by showing that slight improvements to our result would separate NP from NC. |
