Abstract
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.
Highlights
A central question in complexity theory – the P vs L problem – asks if a deterministic Turing machine that runs in polynomial time can accept any language that cannot be acceptedLeibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany37:2 Lower Bounds for Projective Dimension of Graphs by deterministic Turing machines with logarithmic space bound
The latter, recast in the language of circuit complexity theory, asks if there exists an explicit family of functions (f : {0, 1}n → {0, 1}) computable in polynomial time, such that any family of deterministic branching programs computing them has to be of size 2Ω(n)
By observing properties about the measure of projective dimension, choosing a new candidate function1, we demonstrate that the above restriction can help by proving the following quadratic gap between the two measures
Summary
A central question in complexity theory – the P vs L problem – asks if a deterministic Turing machine that runs in polynomial time can accept any language that cannot be accepted. Pudlák and Rödl [12] described a linear algebraic approach to show size lower bounds against deterministic branching programs They introduced a linear algebraic parameter called projective dimension (denoted by pdF(f ), over a field F) defined on a natural graph associated with the Boolean function f. In order to establish size lower bounds against branching programs, it suffices to prove lower bounds for projective dimension of explicit family of Boolean functions. We demonstrate the weakness of this measure by showing the existence of a function ( not explicit) for which there is an exponential gap between upd over any partition and the branching program size (Proposition 5.1) This motivates us to look for variants of projective dimension of graphs, which is closer to the optimal branching program size of the corresponding Boolean function.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
