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Documents authored by Peitl, Tomáš

Document
DOI: 10.4230/LIPIcs.SAT.2023.13
A SAT Solver’s Opinion on the Erdős-Faber-Lovász Conjecture

Authors: Markus Kirchweger, Tomáš Peitl, and Stefan Szeider

Published in: LIPIcs, Volume 271, 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)

Abstract
In 1972, Paul Erdős, Vance Faber, and Lászlo Lovász asked whether every linear hypergraph with n vertices can be edge-colored with n colors, a statement that has come to be known as the EFL conjecture. Erdős himself considered the conjecture as one of his three favorite open problems, and offered increasing money prizes for its solution on several occasions. A proof of the conjecture was recently announced, for all but a finite number of hypergraphs. In this paper we look at some of the cases not covered by this proof. We use SAT solvers, and in particular the SAT Modulo Symmetries (SMS) framework, to generate non-colorable linear hypergraphs with a fixed number of vertices and hyperedges modulo isomorphisms. Since hypergraph colorability is NP-hard, we cannot directly express in a propositional formula that we want only non-colorable hypergraphs. Instead, we use one SAT (SMS) solver to generate candidate hypergraphs modulo isomorphisms, and another to reject them by finding a coloring. Each successive candidate is required to defeat all previous colorings, whereby we avoid having to generate and test all linear hypergraphs. Computational methods have previously been used to verify the EFL conjecture for small hypergraphs. We verify and extend these results to larger values and discuss challenges and directions. Ours is the first computational approach to the EFL conjecture that allows producing independently verifiable, DRAT proofs.
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Markus Kirchweger, Tomáš Peitl, and Stefan Szeider. A SAT Solver’s Opinion on the Erdős-Faber-Lovász Conjecture. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kirchweger_et_al:LIPIcs.SAT.2023.13,
  author =	{Kirchweger, Markus and Peitl, Tom\'{a}\v{s} and Szeider, Stefan},
  title =	{{A SAT Solver’s Opinion on the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture}},
  booktitle =	{26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)},
  pages =	{13:1--13:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-286-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{271},
  editor =	{Mahajan, Meena and Slivovsky, Friedrich},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2023.13},
  URN =		{urn:nbn:de:0030-drops-184752},
  doi =		{10.4230/LIPIcs.SAT.2023.13},
  annote =	{Keywords: hypergraphs, graph coloring, SAT modulo symmetries}
}
Document
DOI: 10.4230/LIPIcs.SAT.2022.11
Should Decisions in QCDCL Follow Prefix Order?

Authors: Benjamin Böhm, Tomáš Peitl, and Olaf Beyersdorff

Published in: LIPIcs, Volume 236, 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)

Abstract
Quantified conflict-driven clause learning (QCDCL) is one of the main solving approaches for quantified Boolean formulas (QBF). One of the differences between QCDCL and propositional CDCL is that QCDCL typically follows the prefix order of the QBF for making decisions. We investigate an alternative model for QCDCL solving where decisions can be made in arbitrary order. The resulting system QCDCL^ANY is still sound and terminating, but does not necessarily allow to always learn asserting clauses or cubes. To address this potential drawback, we additionally introduce two subsystems that guarantee to always learn asserting clauses (QCDCL^UNI-ANI) and asserting cubes (QCDCL^EXI-ANY), respectively. We model all four approaches by formal proof systems and show that QCDCL^UNI-ANY is exponentially better than QCDCL on false formulas, whereas QCDCL^EXI-ANY is exponentially better than QCDCL on true QBFs. Technically, this involves constructing specific QBF families and showing lower and upper bounds in the respective proof systems. We complement our theoretical study with some initial experiments that confirm our theoretical findings.
Cite as

Benjamin Böhm, Tomáš Peitl, and Olaf Beyersdorff. Should Decisions in QCDCL Follow Prefix Order?. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bohm_et_al:LIPIcs.SAT.2022.11,
  author =	{B\"{o}hm, Benjamin and Peitl, Tom\'{a}\v{s} and Beyersdorff, Olaf},
  title =	{{Should Decisions in QCDCL Follow Prefix Order?}},
  booktitle =	{25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-242-6},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{236},
  editor =	{Meel, Kuldeep S. and Strichman, Ofer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2022.11},
  URN =		{urn:nbn:de:0030-drops-166850},
  doi =		{10.4230/LIPIcs.SAT.2022.11},
  annote =	{Keywords: QBF, CDCL, proof complexity, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.12
Hard QBFs for Merge Resolution

Authors: Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl, and Gaurav Sood

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. As demonstrated in [Olaf Beyersdorff et al., 2020], this makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas which are hard for classical QBF resolution systems. Here we show the first exponential lower bounds for MRes, thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems ∀Exp+Res and IR.
Cite as

Olaf Beyersdorff, Joshua Blinkhorn, Meena Mahajan, Tomáš Peitl, and Gaurav Sood. Hard QBFs for Merge Resolution. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{beyersdorff_et_al:LIPIcs.FSTTCS.2020.12,
  author =	{Beyersdorff, Olaf and Blinkhorn, Joshua and Mahajan, Meena and Peitl, Tom\'{a}\v{s} and Sood, Gaurav},
  title =	{{Hard QBFs for Merge Resolution}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.12},
  URN =		{urn:nbn:de:0030-drops-132530},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.12},
  annote =	{Keywords: QBF, resolution, proof complexity, lower bounds}
}
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