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 (Pudl\'ak, R\"odl 1992), who showed that proving lower bounds for pd$(G_f)$ for bipartite graphs $G_f$ associated with a Boolean function $f$ imply size lower bounds for branching programs computing $f$. Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy 2000), 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 $2^{\Omega(n)}$. Motivated by the argument in (Pudl\'ak, R\"odl 1992), 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$(G)$). As our main result, we show that there is an explicit family of graphs on $N = 2^n$ vertices such that the projective dimension is $O(\sqrt{n})$, the projective dimension with intersection dimension $1$ is $\Omega(n)$ and the bitwise decomposable projective dimension is $\Omega(\frac{n^{1.5}}{\log n})$. We also show that there exist a Boolean function $f$ (on $n$ bits) for which the gap between upd$(G_f)$ and bpsize$(f)$ is $2^{\Omega(n)}$. 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 $f$, $\textrm{bitpdim}(G_f)/6 \le \textrm{bpsize}(f) \le (\textrm{bitpdim}(G_f))^c$. 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.Comment: 24 pages, 3 figure

Pebbling, Entropy and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et. al. (2012). Proving a super-polynomial lower bound for the size of nondeterministic thrifty branching programs (NTBP) would separate $NL$ from $P$ for thrifty models solving the tree evaluation problem. First, we show that {\em Read-Once NTBPs} are equivalent to whole black-white pebbling algorithms thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of NTBPs called {\em Bitwise Independence}. The best known NTBPs (of size $O(k^{h/2+1})$) for the tree evaluation problem given by Cook et. al. (2012) are Bitwise Independent. As our main result, we show that any Bitwise Independent NTBP solving $TEP_{2}^{h}(k)$ must have at least $\frac{1}{2}k^{h/2}$ states. Prior to this work, lower bounds were known for NTBPs only for fixed heights $h=2,3,4$ (See Cook et. al. (2012)). We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent NTBP solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Z{\'a}k (2001) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds given by Cook et. al. (2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any $k$-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.Comment: 25 Pages, Manuscript submitted to Journal in June 2013 This version includes a proof for tight size bounds for (syntactic) read-once NTBPs. The proof is in the same spirit as the proof for size bounds for bitwise independent NTBPs present in the earlier version of the paper and is included in the journal version of the paper submitted in June 201

Deterministic Black-Box Identity Testing π\pi-Ordered Algebraic Branching Programs

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. Given a permutation $\pi$ of $n$ variables, for a $\pi$-ordered ABP ($\pi$-OABP), for any directed path $p$ from source to sink, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $\pi$. An ABP $A$ is said to be of read $r$, if any variable appears at most $r$ times in $A$. Our main result pertains to the identity testing problem. Over any field $F$ and in the black-box model, i.e. given only query access to the polynomial, we have the following result: read $r$ $\pi$-OABP computable polynomials can be tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]. Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $\Omega(2^n/n)$ and read $\Omega(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $G$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $\pi \neq \pi'$, such that $p$ can be computed by a read-once $\pi$-OABP, but where any $\pi'$-OABP computing $p$ must read some variable at least $2^n