Decision DNNFs with imbalanced conjunction cannot efficiently represent CNFs of bounded width

TL;DR

This paper investigates the limitations of Decision DNNFs with imbalanced conjunction gates in representing CNFs of bounded primal treewidth, establishing lower bounds on their size and answering an open question negatively.

Contribution

It introduces a lower bound for Decision DNNFs with imbalanced conjunction gates representing CNFs of bounded primal treewidth, advancing understanding of their computational limitations.

Findings

Established a lower bound of n^{Ω((1-α)·k)} for representation size.

Answered negatively an open question about representing bounded primal treewidth CNFs with α-imbalanced Decision DNNFs.

Introduced a novel concept of bidimensionality in parameterized complexity.

Abstract

Decomposable Negation Normal Forms \textsc{dnnf} [Darwiche, 'Decomposable Negation Normal Form', JACM, 2001] is a landmark Knowledge Compilation (\textsc{kc}) model, highly important both in \textsc{ai} and Theoretical Computer Science. Numerous restrictions of the model have been studied. In this paper we consider the restriction where all the gates are $\alpha$-imbalanced that is, at most one input of each gate depends on more than $n^{\alpha}$ variables (where $n$ is the number if variables of the function being represented). The concept of imbalanced gates has been first considered in [Lai, Liu, Yin 'New canonical representations by augmenting OBDDs with conjunctive decomposition', JAIR, 2017]. We consider the idea in the context of representation of \textsc{cnf}s of bounded primal treewidth. We pose an open question as to whether \textsc{cnf}s of bounded primal treewidth can be represented as \textsc{fpt}-sized \textsc{dnnf} with $\alpha$-imbalanced gates. We answer the question negatively for Decision \textsc{dnnf} with $\alpha$-imbalanced conjunction gates. In particular, we establish a lower bound of $n^{\Omega((1-\alpha) \cdot k)}$ for the representation size (where $k$ is the primal treewidth of the input \textsc{cnf}). The main engine for the above lower bound is a combinatorial result that may be of an independent interest in the area of parameterized complexity as it introduces a novel concept of bidimensionality.