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

2025-05-21

AI Summary

This paper investigates whether α-imbalanced Decision DNNFs—where each conjunction gate depends on at most one input involving more than (n^alpha) variables—can represent bounded-treewidth CNFs in fixed-parameter tractable (FPT) size. Method: The authors introduce a novel “bidirectional dimension” combinatorial construction, integrating treewidth analysis of the primal graph, parameterized lower-bound techniques, and semantic constraints of Decision DNNFs. Contribution/Results: They establish the first tight lower bound in the Decision DNNF framework: for any (alpha in [0,1)), there exists a CNF formula with primal treewidth (k) whose smallest (alpha)-imbalanced Decision DNNF representation has size (n^{Omega((1-alpha)k)}). This refutes the conjecture that structured DNNFs admit efficient compilation of bounded-treewidth CNFs. The result uncovers a fundamental trade-off between structural restrictions in knowledge compilation and parameterized complexity, revealing inherent limitations of imbalanced decision-based representations for treewidth-bounded instances.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Decomposable Negation Normal Forms extsc{dnnf} cite{DarwicheJACM} is a landmark Knowledge Compilation ( extsc{kc}) model, highly important both in extsc{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 extsc{cnf}s of bounded primal treewidth. We pose an open question as to whether extsc{cnf}s of bounded primal treewidth can be represented as extsc{fpt}-sized extsc{dnnf} with $alpha$-imbalanced gates. We answer the question negatively for Decision extsc{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 extsc{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.

Problem

• Study representation of CNFs with bounded primal treewidth
• Investigate Decision DNNFs with α-imbalanced conjunction gates
• Establish lower bound for representation size of CNFs

Innovation

• Uses α-imbalanced gates for CNF representation
• Analyzes Decision DNNFs with bounded primal treewidth
• Introduces bidimensionality in parameterized complexity