Remarks on Graph Complexity

We revisit the notion of graph complexity introduced by Pudll ak, RR odl, and Savick y PRS]. Using their framework, we show that suuciently strong superlinear monotone lower bounds for the very special class of 2-slice functions would imply superpolynomial lower bounds for some other functions. Given an n-vertex graph G, the corresponding 2-slice function fG on n variables evaluates to zero on inputs with less than two 1's and evaluates to one on inputs with more than two 1's. On inputs with exactly two 1's, fG evaluates to 1 exactly when the pair of variables set to 1 corresponds to an edge in G. Combining our observations with those from PRS], we can show, for instance, that a lower bound of n 1+(1) on the (monotone) formula size of an explicit 2-slice function f on n variables would imply a 2 (l) lower bound on the formula size of another explicit function g on l variables, where l = (log n). We consider lower bound questions for depth-3 bipartite graph complexity. We prove some weak lower bounds on this measure using algebraic methods. For instance, our results give a lower bound of ((log n= log log n) 2) for bipartite graphs arising from Hadamard matrices, such as the Paley-type bipartite graphs. A lower bound of n (1) on the depth-3 complexity of an explicit bipartite graph would give superlinear size lower bounds on log-depth boolean circuits for an explicit function. Similarly, a lower bound of 2 (log n) (1) would give an explicit language outside the class cc 2 of the two-party communication complexity.