Parameterized Compilation Lower Bounds for Restricted CNF-formulas

TL;DR

This paper establishes fundamental lower bounds on the size of DNNF circuits encoding CNF formulas with certain graph width restrictions, revealing limitations in knowledge compilation.

Contribution

It provides the first unconditional parameterized lower bounds for DNNF encodings of CNF formulas based on graph width measures.

Findings

DNNF size grows as n^{Omega(k)} for formulas with modular incidence treewidth k.

DNNF size grows as n^{Omega(sqrt(k))} for formulas with incidence neighborhood diversity k.

Results show significant differences between these parameters and treewidth in their impact on compilation complexity.

Abstract

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 $n^{\Omega(k)}$, and - there are CNF formulas of size $n$ and incidence neighborhood diversity $k$ whose smallest DNNF-encoding has size $n^{\Omega(\sqrt{k})}$. 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.