Simple minimally unsatisfiable subsets of 2-CNFs

2026-03-11

AI Summary

This study investigates the computational complexity and efficient algorithms for identifying simple minimal unsatisfiable subsets (MUSes) containing unit clauses in 2-CNF formulas. Building upon the Abbasizanjani–Kullmann classification framework and leveraging graph-theoretic and Boolean satisfiability techniques, the work presents the first linear-time algorithm for recognizing 2-MU formulas. It further establishes that MUSes with one or two unit clauses can be decided and constructed in polynomial time. An incremental polynomial-time enumeration algorithm tailored to such MUSes is also developed. By delineating the boundary between tractable and intractable cases of MUSes in 2-CNF, this research offers new insights into the structural properties underlying unsatisfiability in this fragment of propositional logic.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We present a study of minimal unsatisfiable subsets (MUSs) of 2-CNF Boolean formulas, building on the Abbasizanjani-Kullmann classification of minimally unsatisfiable 2-CNFs (2-MUs). We start by giving a linear-time procedure for recognising 2-MUs. Then we study the problem of finding one simple MUS. On the one hand we extend the results by Kleine Buening et al, which showed NP-completeness of the decision, whether a deficiency-1 MUS exists. On the other hand we show that deciding/finding an MUS containing one or two unit-clauses (which are special deficiency-1 MUSs) can be done in polynomial time. Finally we present an incremental polynomial time algorithm for some special type of MUSs, namely those MUSs containing at least one unit-clause. We conclude by discussing the main open problem, developing a deeper understanding of the landscape of easy/hard MUSs of 2-CNFs.

Problem

• minimal unsatisfiable subsets
• 2-CNF
• computational complexity
• unit-clauses
• deficiency

Innovation

• minimal unsatisfiable subsets
• 2-CNF
• polynomial-time algorithm
• unit-clause
• deficiency-1