The splitting power of branching programs of bounded repetition and CNFs of bounded width

In this paper we study syntactic branching programs of bounded repetition representing CNFs of bounded treewidth. For this purpose we introduce two new structural graph parameters $d$-pathwidth and clique preserving $d$-pathwidth denoted by $p{w}_d\left(G\right)$ and $cp{w}_d\left(G\right)$ where $G$ is a graph. We show that $cp{w}_2\left(G\right)\le O\left(tw\left(G\right)\Delta \left(G\right)\right)$ where $tw\left(G\right)$ and $\Delta \left(G\right)$ are, respectively the treewidth and maximal degree of $G$. Using this upper bound, we demonstrate that each CNF $\psi$ can be represented as a conjunction of two OBDDs (quite a restricted class of read-twice branching programs) of size $2^{O\left(\Delta \left(\psi \right)\cdot tw{\left(\psi \right)}^2\right)}$ where $tw\left(\psi \right)$ is the treewidth of the primal graph of $\psi$ and each variable occurs in $\psi$ at most $\Delta \left(\psi \right)$ times.

Next, we use $d$-pathwidth to obtain lower bounds for monotone branching programs. In particular, we consider the monotone version of syntactic nondeterministic read $d$ times branching programs (just forbidding negative literals as edge labels) and introduce a further restriction that each computational path can be partitioned into at most $d$ read-once subpaths. We call the resulting model separable monotone read $d$ times branching programs and abbreviate them $d$-SMNBPs. For each graph $G$ without isolated vertices, we introduce a CNF $\psi \left(G\right)$ whose clauses are $(u\vee e\vee v)$ for each edge $e=\{u,v\}$ of $G$. We prove that a $d$-SMNBP representing $\psi \left(G\right)$ is of size at least $\Omega \left(c^{p{w}_d\left(G\right)}\right)$ where $c={(8/7)}^{1/12}$. We use this ’generic’ lower bound to obtain an exponential lower bound for a ’concrete’ class of CNFs $\psi \left(K_n\right)$. In particular, we demonstrate that for each $0<a<1$, the size of $n^a$-SMNBP representing $\psi \left(K_n\right)$ is at least $c^{n^b}$ where $b$ is an arbitrary constant such that $a+b<1$. This lower bound is tight in the sense $\psi \left(K_n\right)$ can be represented by a poly-sized $n$-SMNBP.