Characterization and Lower Bounds for Branching Program Size Using Projective Dimension

We study projective dimension, a graph parameter (denoted by pd(G) for a graph G), introduced by Pudlák and Rödl [13], who showed that proving lower bounds for pd(Gf ) for bipartite graphs Gf associated with a Boolean function f imply size lower bounds for branching programs computing f . Despite several attempts [13, 17], proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We show that there exist a Boolean function f (on n bits) for which the gap between the projective dimension and size of the optimal branching program computing f (denoted by bpsize(f)), is 2. Motivated by the argument in [13], we define two variants of projective dimension projective dimension with intersection dimension 1 (denoted by upd(G)) and bitwise decomposable projective dimension (denoted by bitpdim(G)). We show the following results : (a) There is an explicit family of graphs on N = 2 vertices such that • the projective dimension is O( √ n). • the projective dimension with intersection dimension 1 is Ω(n). • the bitwise decomposable projective dimension is Ω( n 1.5 log n ) (b) We show that there exist a Boolean function f (on n bits) for which the gap between upd(Gf ) and bpsize(f) is 2 . In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a large constant c > 0 and for any function f , bitpdim(Gf )/6 ≤ bpsize(f) ≤ (bitpdim(Gf )) (c) We also study two other variants of projective dimension and show that they are exactly equal to well-studied graph parameters bipartite clique cover number and bipartite partition number respectively. This immediately yields exponential lower bounds for these measures. ∗Indian Institute of Technology Madras, Chennai, India. ({kdinesh,sajin,jayalal}@cse.iitm.ac.in) 1 ar X iv :1 60 4. 07 20 0v 1 [ cs .C C ] 2 5 A pr 2 01 6