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DOI: 10.4230/LIPIcs.SAT.2026.28

Factoring Learned Clauses

Authors: Florian Pollitt, Zachary Battleman, Mathias Fleury, Yakir Vizel, Marijn J. H. Heule, Armin Biere, and Randal E. Bryant

Published in: LIPIcs, Volume 377, 29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026)

Abstract

Modern SAT solvers are based on the conflict-driven clause learning (CDCL) paradigm, which can be simulated by the resolution proof system. This limits solver effectiveness on instances known to be hard for resolution. Certain approaches, such as parity reasoning, have been shown to be effective in this context, but are hard to integrate with CDCL, in particular, with mainstream proof certificates. The powerful yet simple Extended Resolution (ER) proof system provides an alternative but is not widely used in SAT solving despite having proof certificates for decades and using it effectively remains an open challenge. This paper revisits previous work on ER, which factors out repeated parts of learned clauses during conflict analysis, and explores how their original strategy benefits from 15 years of improvements in the state-of-the-art solver CaDiCaL. We further propose a new, less intrusive inprocessing approach based on factoring XOR and ITE gates from learned clauses globally. Previous work on bounded variable addition focused on AND gates and original clauses only. Our experimental evaluation shows substantial improvements on hard combinatorial benchmark families without performance degradation on the SAT Competition.

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Florian Pollitt, Zachary Battleman, Mathias Fleury, Yakir Vizel, Marijn J. H. Heule, Armin Biere, and Randal E. Bryant. Factoring Learned Clauses. In 29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 377, pp. 28:1-28:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{pollitt_et_al:LIPIcs.SAT.2026.28,
  author =	{Pollitt, Florian and Battleman, Zachary and Fleury, Mathias and Vizel, Yakir and Heule, Marijn J. H. and Biere, Armin and Bryant, Randal E.},
  title =	{{Factoring Learned Clauses}},
  booktitle =	{29th International Conference on Theory and Applications of Satisfiability Testing (SAT 2026)},
  pages =	{28:1--28:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-431-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{377},
  editor =	{Ignatiev, Alexey and Szeider, Stefan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2026.28},
  URN =		{urn:nbn:de:0030-drops-263343},
  doi =		{10.4230/LIPIcs.SAT.2026.28},
  annote =	{Keywords: SAT solving, Extended Resolution, CDCL, Inprocessing}
}

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DOI: 10.4230/LIPIcs.SAT.2025.3

Problem Partitioning via Proof Prefixes

Authors: Zachary Battleman, Joseph E. Reeves, and Marijn J. H. Heule

Published in: LIPIcs, Volume 341, 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)

Abstract

Satisfiability solvers have been instrumental in tackling hard problems, including mathematical challenges that require years of computation. A key obstacle in efficiently solving such problems lies in effectively partitioning them into many, frequently millions of subproblems. Existing automated partitioning techniques, primarily based on lookahead methods, perform well on some instances but fail to generate effective partitions for many others. This paper introduces a powerful partitioning approach that leverages prefixes of proofs derived from conflict-driven clause-learning solvers. This method enables non-experts to harness the power of massively parallel SAT solving for their problems. We also propose a semantically-driven partitioning technique tailored for problems with large cardinality constraints, which frequently arise in optimization tasks. We evaluate our methods on diverse benchmarks, including combinatorial problems and formulas from SAT and MaxSAT competitions. Our results demonstrate that these techniques outperform existing partitioning strategies in many cases, offering improved scalability and efficiency.

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Zachary Battleman, Joseph E. Reeves, and Marijn J. H. Heule. Problem Partitioning via Proof Prefixes. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{battleman_et_al:LIPIcs.SAT.2025.3,
  author =	{Battleman, Zachary and Reeves, Joseph E. and Heule, Marijn J. H.},
  title =	{{Problem Partitioning via Proof Prefixes}},
  booktitle =	{28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)},
  pages =	{3:1--3:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-381-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{341},
  editor =	{Berg, Jeremias and Nordstr\"{o}m, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2025.3},
  URN =		{urn:nbn:de:0030-drops-237378},
  doi =		{10.4230/LIPIcs.SAT.2025.3},
  annote =	{Keywords: Satisfiability solving, parallel computing, problem partitioning}
}

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Factoring Learned Clauses

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Problem Partitioning via Proof Prefixes

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