We study projective dimension, a graph parameter, denoted by pd( G ) for a bipartite graph G , introduced by Pudlák and Rödl (1992). For a Boolean function f (on n bits), Pudlák and Rödl associated a bipartite graph G f and showed that size of the optimal branching program computing f , denoted by bpsize( f ), is at least pd( G f ) (also denoted by pd( f )). Hence, proving lower bounds for pd( f ) implies lower bounds for bpsize( f ). Despite several attempts (Pudlák and Rödl (1992), Rónyai et al. (2000)), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We observe that there exist a Boolean function f for which the gap between the pd( f ) and bpsize( f )) is 2 Ω( n ) . Motivated by the argument in Pudlák and Rödl (1992), we define two variants of projective dimension: projective dimension with intersection dimension 1 , denoted by upd( f ), and bitwise decomposable projective dimension , denoted by bitpdim( f ). We show the following results: (a) We observe that there exist a Boolean function f for which the gap between upd( f ) and bpsize( f ) is 2 Ω( n ) . 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 constant c > 0 and for any function f , bitpdim( f )/6 ≤ bpsize( f ) ≤ (bitpdim( f )) c . (b) We introduce a new candidate family of functions f for showing super-polynomial lower bounds for bitpdim( f ). As our main result, for this family of functions, we demonstrate gaps between pd( f ) and the above two new measures for f : pd( f ) = O (√ n ) upd( f ) = Ω ( n ) bitpdim( f ) = Ω ( n 1.5 / log n ). We adapt Nechiporuk’s techniques for our linear algebraic setting to prove the best-known bpsize lower bounds for bitpdim. Motivated by this linear algebraic setting of our main result, we derive exponential lower bounds for two restricted variants of pd( f ) and upd( f ) by observing that they are exactly equal to well-studied graph parameters—bipartite clique cover number and bipartite partition number, respectively.