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

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.