Theory Comput Syst (2017) 61:755–776
On Oblivious Branching Programs with Bounded
Repetition that Cannot Efficiently Compute CNFs
of Bounded Treewidth
Published online: 17 October 2016
© Springer Science+Business Media New York 2016
Abstract In this paper we study complexity of an extension of ordered binary
decision diagrams (OBDDs) called c-OBDDs on CNFs of bounded (primal graph)
treewidth. In particular, we show that for each k ≥ 3 there is a class of CNFs of
treewidth k for which the equivalent c-OBDDs are of size (n
this lower bound holds if c-OBDDs are non-deterministic and semantic. Our second
result uses the above lower bound to separate the above model from sentential deci-
sion diagrams (SDDs). In order to obtain the lower bound, we use a structural graph
parameter called matching width. Our third result shows that matching width and
pathwidth are linearly related.
Ordered Binary Decision Diagrams (OBDDs) is a famous representation of Boolean
functions being actively investigated from both applied and theoretical perspective.
The theoretical research, among other things, has resulted in many upper and lower
bounds on OBDD size realizing various classes of functions .
One such an upper bound, established in  states that a CNF of treewidth k of
its primal graph can be represented by an OBDD of size O(n
). In terms of param-
eterized complexity, this is an XP upper bound, that is the degree of the polynomial
depends on k. A natural open question is whether this upper bound can be improved
to an FPT upper bound, i.e. one of the form f(k)∗n
,wherec is a universal constant.
Department of Computer Science and Information Systems, Birkbeck, University of London,