Parameterized Compilation Lower Bounds for Restricted CNF-formulas
This provides foundational insights into the complexity of knowledge compilation, showing that certain graph parameters behave differently from treewidth, which is significant for researchers in computational logic and AI.
The paper tackles the problem of knowledge compilation for restricted CNF-formulas by showing unconditional parameterized lower bounds on the size of DNNF circuits, with results including sizes of n^{Ω(k)} for modular incidence treewidth and n^{Ω(√k)} for incidence neighborhood diversity.
We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that - there are CNF formulas of size $n$ and modular incidence treewidth $k$ whose smallest DNNF-encoding has size $n^{Ω(k)}$, and - there are CNF formulas of size $n$ and incidence neighborhood diversity $k$ whose smallest DNNF-encoding has size $n^{Ω(\sqrt{k})}$. These results complement recent upper bounds for compiling CNF into DNNF and strengthen---quantitatively and qualitatively---known conditional low\-er bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth.