Connecting Width and Structure in Knowledge Compilation

Several query evaluation tasks can be done via knowledge compilation: the query result is compiled as a lineage circuit from which the answer can be determined. For such tasks, it is important to leverage some width parameters of the circuit, such as bounded treewidth or pathwidth, to convert the circuit to structured classes, e.g., deterministic structured NNFs (d-SDNNFs) or OBDDs. In this work, we show how to connect the width of circuits to the size of their structured representation, through upper and lower bounds. For the upper bound, we show how bounded-treewidth circuits can be converted to a d-SDNNF, in time linear in the circuit size. Our bound, unlike existing results, is constructive and only singly exponential in the treewidth. We show a related lower bound on monotone DNF or CNF formulas, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable). Specifically, any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth; and the same holds for pathwidth when compiling to OBDDs. Our lower bounds, in contrast with most previous work, apply to any formula of this class, not just a well-chosen family. Hence, for our language of DNF and CNF, pathwidth and treewidth respectively characterize the efficiency of compiling to OBDDs and (d-)SDNNFs, that is, compilation is singly exponential in the width parameter. We conclude by applying our lower bound results to the task of query evaluation

On OBDDs for CNFs of bounded treewidth

In this paper we show that a CNF cannot be compiled into an Ordered Binary Decision Diagram (OBDD) of fixed-parameter size parameterized by the primal graph treewidth of the input CNF. Thus we provide a parameterized separation between OBDDs and Sentential Decision Diagrams for which such fixed-parameter compilation is possible. We also show that the best existing parameterized upper bound for OBDDs in fact holds for the incidence graph treewidth parameterization

On oblivious branching programs with bounded repetition that cannot efficiently compute CNFs of bounded treewidth

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 Ω(nk/(8c−4)). Moreover, 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 decision 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

SDDs are Exponentially More Succinct than OBDDs

Introduced by Darwiche (2011), sentential decision diagrams (SDDs) are essentially as tractable as ordered binary decision diagrams (OBDDs), but tend to be more succinct in practice. This makes SDDs a prominent representation language, with many applications in artificial intelligence and knowledge compilation. We prove that SDDs are more succinct than OBDDs also in theory, by constructing a family of boolean functions where each member has polynomial SDD size but exponential OBDD size. This exponential separation improves a quasipolynomial separation recently established by Razgon (2013), and settles an open problem in knowledge compilation

Classification of OBDD Size for Monotone 2-CNFs

We introduce a new graph parameter called linear upper maximum induced matching width lu-mim width, denoted for a graph G by lu(G). We prove that the smallest size of the obdd for ?, the monotone 2-cnf corresponding to G, is sandwiched between 2^{lu(G)} and n^{O(lu(G))}. The upper bound is based on a combinatorial statement that might be of an independent interest. We show that the bounds in terms of this parameter are best possible. The new parameter is closely related to two existing parameters: linear maximum induced matching width (lmim width) and linear special induced matching width (lsim width). We prove that lu-mim width lies strictly in between these two parameters, being dominated by lsim width and dominating lmim width. We conclude that neither of the two existing parameters can be used instead of lu-mim width to characterize the size of obdds for monotone 2-cnfs and this justifies introduction of the new parameter

No small nondeterministic read-once branching programs for CNFs of bounded treewidth

In this paper, given a parameter k, we demonstrate an infinite class of cnfs of treewidth at most k of their primal graphs such that equivalent nondeterministic read-once branching programs (nrobps) are of size at least nck for some universal constant c. Thus we rule out the possibility of fixed-parameter tractable space complexity of nrobps parameterized by the smallest treewidth of equivalent cnfs