CV-width: A New Complexity Parameter for CNFs

We present new complexity results on the compilation of CNFs into DNNFs and OBDDs. In particular, we introduce a new notion of width, called CV-width, which is specific to CNFs and that dominates the treewidth of the CNF incidence graph. We then show that CNFs can be compiled into structured DNNFs in time and space that are exponential only in CV-width. Not only does CV-width dominate the incidence graph treewidth, but the former width can be bounded when the latter is unbounded. We also introduce a restricted version of CV-width, called linear CV-width, and show that it dominates both pathwidth and cutwidth, which have been used to bound the complexity of OBDDs. We show that CNFs can be compiled into OBDDs in time and space that are exponential only in linear CV-width. We also show that linear CV-width can be bounded when pathwidth and cutwidth are unbounded. The new notion of width significantly improves existing upper bounds on both structured DNNFs and OBDDs, and is motived by a new decomposition technique that combines variable splitting with clause splitting.