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

Abstract: 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 $d-pw(G)$ and $d-cpw(G)$ where $G$ is a graph. We show that $2-cpw(G) \leq O(tw(G) \Delta(G))$ where $tw(G)$ and $\Delta(G)$ 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 of size $2<sup>{O(\Delta(\psi)*tw(\psi)<sup>2)}$ where $tw(\psi)$ is the treewidth of the primal graph of $\psi$ and each variable occurs in $\psi$ at most $\Delta(\psi)$ times. Next we use $d$-pathwdith 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 $\psi(G)$ whsose clauses are $(u \vee e \vee v)$ for each edge $e={u,v}$ of $G$. We prove that a $d$-SMNBP representing $\psi(G)$ is of size at least $\Omega(c<sup>{d-pw(G)})$ where $c=(8/7)<sup>{1/12}$. We use this 'generic' lower bound to obtain an exponential lower bound for a 'concrete' class of CNFs $\psi(K_n)$. In particular, we demonstrate that for each $0<a<1$, the size of $n<sup>{a}$-SMNBP representing $\psi(K_n)$ is at least $c<sup>{n<sup>b}$ where $b$ is an arbitrary constant such that $a+b&lt;1$. This lower bound is tight in the sense $\psi(K_n)$ can be represented by a poly-sized $n$-SMNBP.