Characterization and Lower Bounds for Branching Program Size using Projective Dimension

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 Gf and showed that size of the optimal branching program computing f, denoted by bpsize(f), is at least pd(Gf) (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 2Ω(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 2Ω(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 (n1.5 / 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.