On Compiling Structured CNFs to OBDDs

TL;DR

This paper investigates the size of Ordered Binary Decision Diagrams (OBDDs) for structured classes of CNF formulas, establishing conditions for polynomial size and proving exponential lower bounds for certain graph classes.

Contribution

It introduces the few subterms property as a sufficient condition for polynomial OBDD size and characterizes classes of CNFs with variable convex incidence graphs.

Findings

CNFs with variable convex incidence graphs have polynomial OBDD size.

Classes of CNFs with bounded treewidth also have polynomial OBDD size.

There is an exponential lower bound on OBDD size for CNFs with bounded degree incidence graphs.

Abstract

We present new results on the size of OBDD representations of structurally characterized classes of CNF formulas. First, we identify a natural sufficient condition, which we call the few subterms property, for a class of CNFs to have polynomial OBDD size; we then prove that CNFs whose incidence graphs are variable convex have few subterms (and hence have polynomial OBDD size), and observe that the few subterms property also explains the known fact that classes of CNFs of bounded treewidth have polynomial OBDD size. Second, we prove an exponential lower bound on the OBDD size of a family of CNF classes with incidence graphs of bounded degree, exploiting the combinatorial properties of expander graphs.