Simple minimally unsatisfiable subsets of 2-CNFs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917333177466880 |
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| author | Kullmann, Oliver Clewer, Edward |
| author_facet | Kullmann, Oliver Clewer, Edward |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10944 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simple minimally unsatisfiable subsets of 2-CNFs Kullmann, Oliver Clewer, Edward Data Structures and Algorithms 68Q25 F.2.2 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. |
| title | Simple minimally unsatisfiable subsets of 2-CNFs |
| topic | Data Structures and Algorithms 68Q25 F.2.2 |
| url | https://arxiv.org/abs/2603.10944 |