The splitting power of branching programs of bounded repetition and CNFs of bounded width
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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) ≤ O(tw(G) Δ(G)) where tw(G) and Δ(G) are, respectively the treewidth and maximal degree of G. Using this upper bound, we demonstrate that each CNF ψ can be represented as a conjunction of two OBDDs of size 2^O(Δ(ψ)*tw(ψ)^2) where tw(ψ) is the treewidth of the primal graph of ψ and each variable occurs in ψ at most Δ(ψ) 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 ψ(G) whsose clauses are (u ∨ e ∨ v) for each edge e={u,v} of G. We prove that a d-SMNBP representing ψ(G) is of size at least Ω(c^d-pw(G)) where c=(8/7)^1/12. We use this 'generic' lower bound to obtain an exponential lower bound for a 'concrete' class of CNFs ψ(K_n). In particular, we demonstrate that for each 0<a<1, the size of n^a-SMNBP representing ψ(K_n) 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 ψ(K_n) can be represented by a poly-sized n-SMNBP.
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