For CNFs that encode a ternary tree crossed with a path, Decision DNNFs whose conjunction gates split variables imbalancedly require size at least n^{Ω((1−α)√k)}, ruling out FPT-sized representations.
Connecting knowledge compilation classes and width parameters
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Decision DNNFs with imbalanced conjunction cannot efficiently represent CNFs of bounded width
For CNFs that encode a ternary tree crossed with a path, Decision DNNFs whose conjunction gates split variables imbalancedly require size at least n^{Ω((1−α)√k)}, ruling out FPT-sized representations.