Document

**Published in:** LIPIcs, Volume 307, 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)

The enhanced performance of today’s MaxSAT solvers has elevated their appeal for many large-scale applications, notably in software analysis and computer-aided design. Our research delves into refining anytime MaxSAT solving by repeatedly identifying and solving with an exact solver smaller subinstances that are chosen based on the graphical structure of the instance. We investigate various strategies to pinpoint these subinstances. This structure-guided selection of subinstances provides an exact solver with a high potential for improving the current solution. Our exhaustive experimental analyses contrast our methodology as instantiated in our tool MaxSLIM with previous studies and benchmark it against leading-edge MaxSAT solvers.

André Schidler and Stefan Szeider. Structure-Guided Local Improvement for Maximum Satisfiability. In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 26:1-26:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{schidler_et_al:LIPIcs.CP.2024.26, author = {Schidler, Andr\'{e} and Szeider, Stefan}, title = {{Structure-Guided Local Improvement for Maximum Satisfiability}}, booktitle = {30th International Conference on Principles and Practice of Constraint Programming (CP 2024)}, pages = {26:1--26:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-336-2}, ISSN = {1868-8969}, year = {2024}, volume = {307}, editor = {Shaw, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2024.26}, URN = {urn:nbn:de:0030-drops-207112}, doi = {10.4230/LIPIcs.CP.2024.26}, annote = {Keywords: maximum satisfiability, large neighborhood search (LNS), SAT-based local improvement (SLIM), incomplete MaxSAT, graphical structure, metaheuristic} }

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Short Paper

**Published in:** LIPIcs, Volume 307, 30th International Conference on Principles and Practice of Constraint Programming (CP 2024)

Envy-freeness up to any good (EFX) is a key concept in Computational Social Choice for the fair division of indivisible goods, where no agent envies another’s allocation after removing any single item. A deeper understanding of EFX allocations is facilitated by exploring the rainbow cycle number (R_f(d)), the largest number of independent sets in a certain class of directed graphs. Upper bounds on R_f(d) provide guarantees to the feasibility of EFX allocations (Chaudhury et al., EC 2021).
In this work, we precisely compute the numbers R_f(d) for small values of d, employing the SAT modulo Symmetries framework (Kirchweger and Szeider, CP 2021). SAT modulo Symmetries is tailored specifically for the constraint-based isomorph-free generation of combinatorial structures. We provide an efficient encoding for the rainbow cycle number, comparing eager and lazy approaches. To cope with the huge search space, we extend the encoding with invariant pruning, a new method that significantly speeds up computation.

Markus Kirchweger and Stefan Szeider. Computing Small Rainbow Cycle Numbers with SAT Modulo Symmetries (Short Paper). In 30th International Conference on Principles and Practice of Constraint Programming (CP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 307, pp. 37:1-37:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kirchweger_et_al:LIPIcs.CP.2024.37, author = {Kirchweger, Markus and Szeider, Stefan}, title = {{Computing Small Rainbow Cycle Numbers with SAT Modulo Symmetries}}, booktitle = {30th International Conference on Principles and Practice of Constraint Programming (CP 2024)}, pages = {37:1--37:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-336-2}, ISSN = {1868-8969}, year = {2024}, volume = {307}, editor = {Shaw, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2024.37}, URN = {urn:nbn:de:0030-drops-207221}, doi = {10.4230/LIPIcs.CP.2024.37}, annote = {Keywords: EFX, rainbow cycle number, SAT modulo Symmetries, combinatorial search} }

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**Published in:** LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)

eSLIM is a tool for circuit minimization that utilizes Exact Synthesis and the SAT-based local improvement method (SLIM) to locally improve circuits. eSLIM improves upon the earlier prototype CIOPS that uses Quantified Boolean Formulas (QBF) to succinctly encode resynthesis of multi-output subcircuits subject to don't cares. This paper describes two improvements. First, it presents a purely propositional encoding based on a Boolean relation characterizing the input-output behavior of the subcircuit under don't cares. This allows the use of a SAT solver for resynthesis, substantially reducing running times when applied to functions from the IWLS 2023 competition, where eSLIM placed second. Second, it proposes circuit partitioning techniques in which don't cares for a subcircuit are captured only with respect to an enclosing window, rather than the entire circuit. Circuit partitioning trades completeness for efficiency, and successfully enables the application of exact synthesis to some of the largest circuits in the EPFL suite, leading to improvements over the current best implementation for several instances.

Franz-Xaver Reichl, Friedrich Slivovsky, and Stefan Szeider. eSLIM: Circuit Minimization with SAT Based Local Improvement. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{reichl_et_al:LIPIcs.SAT.2024.23, author = {Reichl, Franz-Xaver and Slivovsky, Friedrich and Szeider, Stefan}, title = {{eSLIM: Circuit Minimization with SAT Based Local Improvement}}, booktitle = {27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-334-8}, ISSN = {1868-8969}, year = {2024}, volume = {305}, editor = {Chakraborty, Supratik and Jiang, Jie-Hong Roland}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.23}, URN = {urn:nbn:de:0030-drops-205458}, doi = {10.4230/LIPIcs.SAT.2024.23}, annote = {Keywords: QBF, Exact Synthesis, Circuit Minimization, SLIM} }

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**Published in:** LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)

We obtain the smallest unsatisfiable formulas in subclasses of k-CNF (exactly k distinct literals per clause) with bounded variable or literal occurrences. Smaller unsatisfiable formulas of this type translate into stronger inapproximability results for MaxSAT in the considered formula class. Our results cover subclasses of 3-CNF and 4-CNF; in all subclasses of 3-CNF we considered we were able to determine the smallest size of an unsatisfiable formula; in the case of 4-CNF with at most 5 occurrences per variable we decreased the size of the smallest known unsatisfiable formula. Our methods combine theoretical arguments and symmetry-breaking exhaustive search based on SAT Modulo Symmetries (SMS), a recent framework for isomorph-free SAT-based graph generation. To this end, and as a standalone result of independent interest, we show how to encode formulas as graphs efficiently for SMS.

Tianwei Zhang, Tomáš Peitl, and Stefan Szeider. Small Unsatisfiable k-CNFs with Bounded Literal Occurrence. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{zhang_et_al:LIPIcs.SAT.2024.31, author = {Zhang, Tianwei and Peitl, Tom\'{a}\v{s} and Szeider, Stefan}, title = {{Small Unsatisfiable k-CNFs with Bounded Literal Occurrence}}, booktitle = {27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)}, pages = {31:1--31:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-334-8}, ISSN = {1868-8969}, year = {2024}, volume = {305}, editor = {Chakraborty, Supratik and Jiang, Jie-Hong Roland}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.31}, URN = {urn:nbn:de:0030-drops-205531}, doi = {10.4230/LIPIcs.SAT.2024.31}, annote = {Keywords: k-CNF, (k,s)-SAT, minimally unsatisfiable formulas, symmetry breaking} }

Document

**Published in:** Dagstuhl Reports, Volume 13, Issue 6 (2024)

This report documents the program and the outcomes of Dagstuhl Seminar 23261 "SAT Encodings and Beyond." The seminar facilitated an intense examination and discussion of current results and challenges related to encodings for SAT and related solving paradigms. The seminar featured presentations and group work that provided theoretical, practical, and industrial viewpoints. The goal was to foster more profound insights and advancements in encoding techniques, which are pivotal in enhancing solvers' efficiency.

Marijn J. H. Heule, Inês Lynce, Stefan Szeider, and Andre Schidler. SAT Encodings and Beyond (Dagstuhl Seminar 23261). In Dagstuhl Reports, Volume 13, Issue 6, pp. 106-122, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{heule_et_al:DagRep.13.6.106, author = {Heule, Marijn J. H. and Lynce, In\^{e}s and Szeider, Stefan and Schidler, Andre}, title = {{SAT Encodings and Beyond (Dagstuhl Seminar 23261)}}, pages = {106--122}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2024}, volume = {13}, number = {6}, editor = {Heule, Marijn J. H. and Lynce, In\^{e}s and Szeider, Stefan and Schidler, Andre}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.13.6.106}, URN = {urn:nbn:de:0030-drops-196409}, doi = {10.4230/DagRep.13.6.106}, annote = {Keywords: constraint propagation, lower and upper bounds, problem formulation, propositional satisfiability, symmetry breaking} }

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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

Several works have recently investigated the parameterized complexity of data completion problems, motivated by their applications in machine learning, and clustering in particular. Interestingly, these problems can be equivalently formulated as classical graph problems on induced subgraphs of powers of partially-defined hypercubes.
In this paper, we follow up on this recent direction by investigating the Independent Set problem on this graph class, which has been studied in the data science setting under the name Diversity. We obtain a comprehensive picture of the problem’s parameterized complexity and establish its fixed-parameter tractability w.r.t. the solution size plus the power of the hypercube.
Given that several such FO-definable problems have been shown to be fixed-parameter tractable on the considered graph class, one may ask whether fixed-parameter tractability could be extended to capture all FO-definable problems. We answer this question in the negative by showing that FO model checking on induced subgraphs of hypercubes is as difficult as FO model checking on general graphs.

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2023.16, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan}, title = {{From Data Completion to Problems on Hypercubes: A Parameterized Analysis of the Independent Set Problem}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {16:1--16:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.16}, URN = {urn:nbn:de:0030-drops-194357}, doi = {10.4230/LIPIcs.IPEC.2023.16}, annote = {Keywords: Independent Set, Powers of Hypercubes, Diversity, Parameterized Complexity, Incomplete Data} }

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**Published in:** LIPIcs, Volume 280, 29th International Conference on Principles and Practice of Constraint Programming (CP 2023)

A graph is induced k-universal if it contains all graphs of order k as an induced subgraph. For over half a century, the question of determining smallest k-universal graphs has been studied. A related question asks for a smallest k-universal tournament containing all tournaments of order k.
This paper proposes and compares SAT-based methods for answering these questions exactly for small values of k. Our methods scale to values for which a generate-and-test approach isn't feasible; for instance, we show that an induced 7-universal graph has more than 16 vertices, whereas the number of all connected graphs on 16 vertices, modulo isomorphism, is a number with 23 decimal digits Our methods include static and dynamic symmetry breaking and lazy encodings, employing external subgraph isomorphism testing.

Tianwei Zhang and Stefan Szeider. Searching for Smallest Universal Graphs and Tournaments with SAT. In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{zhang_et_al:LIPIcs.CP.2023.39, author = {Zhang, Tianwei and Szeider, Stefan}, title = {{Searching for Smallest Universal Graphs and Tournaments with SAT}}, booktitle = {29th International Conference on Principles and Practice of Constraint Programming (CP 2023)}, pages = {39:1--39:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-300-3}, ISSN = {1868-8969}, year = {2023}, volume = {280}, editor = {Yap, Roland H. C.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2023.39}, URN = {urn:nbn:de:0030-drops-190760}, doi = {10.4230/LIPIcs.CP.2023.39}, annote = {Keywords: Constrained-based combinatorics, synthesis problems, symmetry breaking, SAT solving, subgraph isomorphism, tournament, directed graphs} }

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Short Paper

**Published in:** LIPIcs, Volume 280, 29th International Conference on Principles and Practice of Constraint Programming (CP 2023)

Motivated by applications from agronomic field experiments, Díaz, Le Bras, and Gomes [CPAIOR 2015] introduced Partially Balanced Latin Rectangles as a generalization of Spatially Balanced Latin Squares. They observed that the generation of Latin rectangles that are optimally balanced is a highly challenging computational problem. They computed, utilizing CSP and MIP encodings, Latin rectangles up to 12 × 12, some optimally balanced, some suboptimally balanced.
In this paper, we develop a SAT encoding for generating balanced Latin rectangles. We compare experimentally encoding variants. Our results indicate that SAT encodings perform competitively with the MIP encoding, in some cases better. In some cases we could find Latin rectangles that are more balanced than previously known ones. This finding is significant, as there are many arithmetic constraints involved. The SAT approach offers the advantage that we can certify that Latin rectangles are optimally balanced through DRAT proofs that can be verified independently.

Vaidyanathan Peruvemba Ramaswamy and Stefan Szeider. Proven Optimally-Balanced Latin Rectangles with SAT (Short Paper). In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 48:1-48:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{peruvembaramaswamy_et_al:LIPIcs.CP.2023.48, author = {Peruvemba Ramaswamy, Vaidyanathan and Szeider, Stefan}, title = {{Proven Optimally-Balanced Latin Rectangles with SAT}}, booktitle = {29th International Conference on Principles and Practice of Constraint Programming (CP 2023)}, pages = {48:1--48:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-300-3}, ISSN = {1868-8969}, year = {2023}, volume = {280}, editor = {Yap, Roland H. C.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2023.48}, URN = {urn:nbn:de:0030-drops-190855}, doi = {10.4230/LIPIcs.CP.2023.48}, annote = {Keywords: combinatorial design, SAT encodings, certified optimality, arithmetic constraints, spatially balanced Latin rectangles} }

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

Modern SAT solvers are frequently embedded as sub-reasoning engines into more complex tools for addressing problems beyond the Boolean satisfiability problem. Examples include solvers for Satisfiability Modulo Theories (SMT), combinatorial optimization, model enumeration and counting. In such use cases, the SAT solver is often able to provide relevant information beyond the satisfiability answer. Further, domain knowledge of the embedding system (e.g., symmetry properties or theory axioms) can be beneficial for the CDCL search, but cannot be efficiently represented in clausal form. In this paper, we propose a general interface to inspect and influence the internal behaviour of CDCL SAT solvers. Our goal is to capture the most essential functionalities that are sufficient to simplify and improve use cases that require a more fine-grained interaction with the SAT solver than provided via the standard IPASIR interface. For our experiments, we extend CaDiCaL with our interface and evaluate it on two representative use cases: enumerating graphs within the SAT modulo Symmetries framework (SMS), and as the main CDCL(T) SAT engine of the SMT solver cvc5.

Katalin Fazekas, Aina Niemetz, Mathias Preiner, Markus Kirchweger, Stefan Szeider, and Armin Biere. IPASIR-UP: User Propagators for CDCL. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{fazekas_et_al:LIPIcs.SAT.2023.8, author = {Fazekas, Katalin and Niemetz, Aina and Preiner, Mathias and Kirchweger, Markus and Szeider, Stefan and Biere, Armin}, title = {{IPASIR-UP: User Propagators for CDCL}}, booktitle = {26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)}, pages = {8:1--8:13}, 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.2023.8}, URN = {urn:nbn:de:0030-drops-184709}, doi = {10.4230/LIPIcs.SAT.2023.8}, annote = {Keywords: SAT, CDCL, Satisfiability Modulo Theories, Satisfiability Modulo Symmetries} }

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

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.

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} }

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

To test a graph’s planarity in SAT-based graph generation we develop SAT encodings with dynamic symmetry breaking as facilitated in the SAT modulo Symmetry (SMS) framework. We implement and compare encodings based on three planarity criteria. In particular, we consider two eager encodings utilizing order-based and universal-set-based planarity criteria, and a lazy encoding based on Kuratowski’s theorem. The performance and scalability of these encodings are compared on two prominent problems from combinatorics: the computation of planar Turán numbers and the Earth-Moon problem. We further showcase the power of SMS equipped with a planarity encoding by verifying and extending several integer sequences from the Online Encyclopedia of Integer Sequences (OEIS) related to planar graph enumeration. Furthermore, we extend the SMS framework to directed graphs which might be of independent interest.

Markus Kirchweger, Manfred Scheucher, and Stefan Szeider. SAT-Based Generation of Planar Graphs. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kirchweger_et_al:LIPIcs.SAT.2023.14, author = {Kirchweger, Markus and Scheucher, Manfred and Szeider, Stefan}, title = {{SAT-Based Generation of Planar Graphs}}, booktitle = {26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023)}, pages = {14:1--14:18}, 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.2023.14}, URN = {urn:nbn:de:0030-drops-184767}, doi = {10.4230/LIPIcs.SAT.2023.14}, annote = {Keywords: SAT modulo Symmetry (SMS), dynamic symmetry breaking, planarity test, universal point set, order dimension, Schnyder’s theorem, Kuratowski’s theorem, Tur\'{a}n’s theorem, Earth-Moon problem} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. Bounded backdoor size implies bounded backdoor depth, but there are formulas of constant backdoor depth and arbitrarily large backdoor size.
We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. We base our FPT approximation algorithm on a sophisticated notion of obstructions, extending Mählmann et al.’s obstruction trees in various ways, including the addition of separator obstructions. We develop the algorithm through a new game-theoretic framework that simplifies the reasoning about backdoors.
Finally, we show that bounded backdoor depth captures tractable classes of CNF formulas not captured by any known method.

Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. SAT Backdoors: Depth Beats Size. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 46:1-46:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dreier_et_al:LIPIcs.ESA.2022.46, author = {Dreier, Jan and Ordyniak, Sebastian and Szeider, Stefan}, title = {{SAT Backdoors: Depth Beats Size}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {46:1--46:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.46}, URN = {urn:nbn:de:0030-drops-169840}, doi = {10.4230/LIPIcs.ESA.2022.46}, annote = {Keywords: satisfiability, backdoor (depth)} }

Document

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We study two variants of the fundamental problem of finding a cluster in incomplete data. In the problems under consideration, we are given a multiset of incomplete d-dimensional vectors over the binary domain and integers k and r, and the goal is to complete the missing vector entries so that the multiset of complete vectors either contains (i) a cluster of k vectors of radius at most r, or (ii) a cluster of k vectors of diameter at most r. We give tight characterizations of the parameterized complexity of the problems under consideration with respect to the parameters k, r, and a third parameter that captures the missing vector entries.

Eduard Eiben, Robert Ganian, Iyad Kanj, Sebastian Ordyniak, and Stefan Szeider. Finding a Cluster in Incomplete Data. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{eiben_et_al:LIPIcs.ESA.2022.47, author = {Eiben, Eduard and Ganian, Robert and Kanj, Iyad and Ordyniak, Sebastian and Szeider, Stefan}, title = {{Finding a Cluster in Incomplete Data}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {47:1--47:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.47}, URN = {urn:nbn:de:0030-drops-169858}, doi = {10.4230/LIPIcs.ESA.2022.47}, annote = {Keywords: Parameterized complexity, incomplete data, clustering} }

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Complete Volume

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

LIPIcs, Volume 241, MFCS 2022, Complete Volume

47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 1-1236, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{szeider_et_al:LIPIcs.MFCS.2022, title = {{LIPIcs, Volume 241, MFCS 2022, Complete Volume}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {1--1236}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022}, URN = {urn:nbn:de:0030-drops-167975}, doi = {10.4230/LIPIcs.MFCS.2022}, annote = {Keywords: LIPIcs, Volume 241, MFCS 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Front Matter, Table of Contents, Preface, Conference Organization

47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{szeider_et_al:LIPIcs.MFCS.2022.0, author = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.0}, URN = {urn:nbn:de:0030-drops-167981}, doi = {10.4230/LIPIcs.MFCS.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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

The SAT modulo Symmetries (SMS) is a recently introduced framework for dynamic symmetry breaking in SAT instances. It combines a CDCL SAT solver with an external lexicographic minimality checking algorithm.
We extend SMS from graphs to matroids and use it to progress on Rota’s Basis Conjecture (1989), which states that one can always decompose a collection of r disjoint bases of a rank r matroid into r disjoint rainbow bases. Through SMS, we establish that the conjecture holds for all matroids of rank 4 and certain special cases of matroids of rank 5. Furthermore, we extend SMS with the facility to produce DRAT proofs. External tools can then be used to verify the validity of additional axioms produced by the lexicographic minimality check.
As a byproduct, we have utilized our framework to enumerate matroids modulo isomorphism and to support the investigation of various other problems on matroids.

Markus Kirchweger, Manfred Scheucher, and Stefan Szeider. A SAT Attack on Rota’s Basis Conjecture. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kirchweger_et_al:LIPIcs.SAT.2022.4, author = {Kirchweger, Markus and Scheucher, Manfred and Szeider, Stefan}, title = {{A SAT Attack on Rota’s Basis Conjecture}}, booktitle = {25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)}, pages = {4:1--4:18}, 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.2022.4}, URN = {urn:nbn:de:0030-drops-166780}, doi = {10.4230/LIPIcs.SAT.2022.4}, annote = {Keywords: SAT modulo Symmetry (SMS), dynamic symmetry breaking, Rota’s basis conjecture, matroid} }

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

Bonnet et al. (FOCS 2020) introduced the graph invariant twin-width and showed that many NP-hard problems are tractable for graphs of bounded twin-width, generalizing similar results for other width measures, including treewidth and clique-width. In this paper, we investigate the use of twin-width for solving the propositional satisfiability problem (SAT) and propositional model counting. We particularly focus on Bounded-ones Weighted Model Counting (BWMC), which takes as input a CNF formula F along with a bound k and asks for the weighted sum of all models with at most k positive literals. BWMC generalizes not only SAT but also (weighted) model counting.
We develop the notion of "signed" twin-width of CNF formulas and establish that BWMC is fixed-parameter tractable when parameterized by the certified signed twin-width of F plus k. We show that this result is tight: it is neither possible to drop the bound k nor use the vanilla twin-width instead if one wishes to retain fixed-parameter tractability, even for the easier problem SAT. Our theoretical results are complemented with an empirical evaluation and comparison of signed twin-width on various classes of CNF formulas.

Robert Ganian, Filip Pokrývka, André Schidler, Kirill Simonov, and Stefan Szeider. Weighted Model Counting with Twin-Width. In 25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 236, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ganian_et_al:LIPIcs.SAT.2022.15, author = {Ganian, Robert and Pokr\'{y}vka, Filip and Schidler, Andr\'{e} and Simonov, Kirill and Szeider, Stefan}, title = {{Weighted Model Counting with Twin-Width}}, booktitle = {25th International Conference on Theory and Applications of Satisfiability Testing (SAT 2022)}, pages = {15:1--15:17}, 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.2022.15}, URN = {urn:nbn:de:0030-drops-166896}, doi = {10.4230/LIPIcs.SAT.2022.15}, annote = {Keywords: Weighted model counting, twin-width, parameterized complexity, SAT} }

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**Published in:** LIPIcs, Volume 235, 28th International Conference on Principles and Practice of Constraint Programming (CP 2022)

The constraint satisfaction problem (CSP) is among the most studied computational problems. While NP-hard, many tractable subproblems have been identified (Bulatov 2017, Zuk 2017). Backdoors, introduced by Williams, Gomes, and Selman (2003), gradually extend such a tractable class to all CSP instances of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a CSP instance and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021) for SAT, is a more refined distance measure, which admits the parallel utilization of different backdoor variables. Bounded backdoor size implies bounded backdoor depth, but there are instances of constant backdoor depth and arbitrarily large backdoor size. Dreier, Ordyniak, and Szeider (2022) provided fixed-parameter algorithms for finding backdoors of small depth into the classes of Horn and Krom formulas.
In this paper, we consider backdoor depth for CSP. We consider backdoors w.r.t. tractable subproblems C_Γ of the CSP defined by a constraint language Γ, i.e., where all the constraints use relations from the language Γ. Building upon Dreier et al.’s game-theoretic approach and their notion of separator obstructions, we show that for any finite, tractable, semi-conservative constraint language Γ, the CSP is fixed-parameter tractable parameterized by the backdoor depth into C_Γ plus the domain size.
With backdoors of low depth, we reach classes of instances that require backdoors of arbitrary large size. Hence, our results strictly generalize several known results for CSP that are based on backdoor size.

Jan Dreier, Sebastian Ordyniak, and Stefan Szeider. CSP Beyond Tractable Constraint Languages. In 28th International Conference on Principles and Practice of Constraint Programming (CP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 235, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dreier_et_al:LIPIcs.CP.2022.20, author = {Dreier, Jan and Ordyniak, Sebastian and Szeider, Stefan}, title = {{CSP Beyond Tractable Constraint Languages}}, booktitle = {28th International Conference on Principles and Practice of Constraint Programming (CP 2022)}, pages = {20:1--20:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-240-2}, ISSN = {1868-8969}, year = {2022}, volume = {235}, editor = {Solnon, Christine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2022.20}, URN = {urn:nbn:de:0030-drops-166490}, doi = {10.4230/LIPIcs.CP.2022.20}, annote = {Keywords: CSP, backdoor depth, constraint language, tractable class, recursive backdoor} }

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**Published in:** LIPIcs, Volume 210, 27th International Conference on Principles and Practice of Constraint Programming (CP 2021)

We propose a novel constraint-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality w.r.t. a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator can perform a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of girth and diameter. With our approach, we could confirm the Simon-Murty Conjecture (1979) on diameter-2-critical graphs for graphs up to 18 vertices.

Markus Kirchweger and Stefan Szeider. SAT Modulo Symmetries for Graph Generation. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kirchweger_et_al:LIPIcs.CP.2021.34, author = {Kirchweger, Markus and Szeider, Stefan}, title = {{SAT Modulo Symmetries for Graph Generation}}, booktitle = {27th International Conference on Principles and Practice of Constraint Programming (CP 2021)}, pages = {34:1--34:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-211-2}, ISSN = {1868-8969}, year = {2021}, volume = {210}, editor = {Michel, Laurent D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2021.34}, URN = {urn:nbn:de:0030-drops-153257}, doi = {10.4230/LIPIcs.CP.2021.34}, annote = {Keywords: symmetry breaking, SAT encodings, graph generation, combinatorial search, extremal graphs, CDCL} }

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**Published in:** Dagstuhl Reports, Volume 7, Issue 9 (2018)

Knowledge compilation (KC) is a research topic which aims to investigate the possibility of circumventing the computational intractability of hard tasks, by preprocessing part of the available information, common to a number of instances. Pioneered almost three decades ago, KC is nowadays a very active research field, transversal to several areas within computer science. Among others, KC intersects knowledge representation, constraint satisfaction, algorithms, complexity theory, machine learning, and databases.
The results obtained so far take various forms, from theory (compilability settings, definition of target languages for KC, complexity results, succinctness results, etc.) to more practical results (development and evaluation of compilers and other preprocessors, applications to diagnosis, planning, automatic configuration, etc.). Recently, KC has been positioned as providing a systematic method for solving problems beyond NP, and also found applications in machine learning.
The goal of this Dagstuhl Seminar was to advance both aspects of KC, and to pave the way for a fruitful cross-fertilization between the topics, from theory to practice. The program included a mixture of long and short presentations, with discussions. Several long talks with a tutorial flavor introduced the participants to the variety of aspects in knowledge compilation and the diversity of techniques used. System presentations as well as an open problem session were also included in the program.

Adnan Darwiche, Pierre Marquis, Dan Suciu, and Stefan Szeider. Recent Trends in Knowledge Compilation (Dagstuhl Seminar 17381). In Dagstuhl Reports, Volume 7, Issue 9, pp. 62-85, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{darwiche_et_al:DagRep.7.9.62, author = {Darwiche, Adnan and Marquis, Pierre and Suciu, Dan and Szeider, Stefan}, title = {{Recent Trends in Knowledge Compilation (Dagstuhl Seminar 17381)}}, pages = {62--85}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {7}, number = {9}, editor = {Darwiche, Adnan and Marquis, Pierre and Suciu, Dan and Szeider, Stefan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.9.62}, URN = {urn:nbn:de:0030-drops-85896}, doi = {10.4230/DagRep.7.9.62}, annote = {Keywords: Knowledge compilation, Constraints, Preprocessing, Probabilistic databases, Model counting} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We show that CSP is fixed-parameter tractable when parameterized by the treewidth of a backdoor into any tractable CSP problem over a finite constraint language. This result combines the two prominent approaches for achieving tractability for CSP: (i) structural restrictions on the interaction between the variables and the constraints and (ii) language restrictions on the relations that can be used inside the constraints. Apart from defining the notion of backdoor-treewidth and showing how backdoors of small treewidth can be used to efficiently solve CSP, our main technical contribution is a fixed-parameter algorithm that finds a backdoor of small treewidth.

Robert Ganian, M. S. Ramanujan, and Stefan Szeider. Combining Treewidth and Backdoors for CSP. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2017.36, author = {Ganian, Robert and Ramanujan, M. S. and Szeider, Stefan}, title = {{Combining Treewidth and Backdoors for CSP}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {36:1--36:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.36}, URN = {urn:nbn:de:0030-drops-69986}, doi = {10.4230/LIPIcs.STACS.2017.36}, annote = {Keywords: Algorithms and data structures, Fixed Parameter Tractability, Constraint Satisfaction} }

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**Published in:** Dagstuhl Follow-Ups, Volume 7, The Constraint Satisfaction Problem: Complexity and Approximability (2017)

A backdoor set of a CSP instance is a set of variables whose instantiation moves the instance into a fixed class of tractable instances (an island of tractability). An interesting algorithmic task is to find a small backdoor set efficiently: once it is found we can solve the instance by solving a number of tractable instances. Parameterized complexity provides an adequate framework for studying and solving this algorithmic task, where the size of the backdoor set provides a natural parameter. In this survey we present some recent parameterized complexity results on CSP backdoor sets, focusing on backdoor sets into islands of tractability that are defined in terms of constraint languages.

Serge Gaspers, Sebastian Ordyniak, and Stefan Szeider. Backdoor Sets for CSP. In The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-Ups, Volume 7, pp. 137-157, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InCollection{gaspers_et_al:DFU.Vol7.15301.137, author = {Gaspers, Serge and Ordyniak, Sebastian and Szeider, Stefan}, title = {{Backdoor Sets for CSP}}, booktitle = {The Constraint Satisfaction Problem: Complexity and Approximability}, pages = {137--157}, series = {Dagstuhl Follow-Ups}, ISBN = {978-3-95977-003-3}, ISSN = {1868-8977}, year = {2017}, volume = {7}, editor = {Krokhin, Andrei and Zivny, Stanislav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DFU.Vol7.15301.137}, URN = {urn:nbn:de:0030-drops-69626}, doi = {10.4230/DFU.Vol7.15301.137}, annote = {Keywords: Backdoor sets, Constraint satisfaction problems, Parameterized complexity, Polymorphisms} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Impagliazzo et al. proposed a framework, based on the logic fragment defining the complexity class SNP, to identify problems that are equivalent to k-CNF-Sat modulo subexponential-time reducibility (serf-reducibility). The subexponential-time solvability of any of these problems implies the failure of the Exponential Time Hypothesis (ETH). In this paper, we extend the framework of Impagliazzo et al., and identify a larger set of problems that are equivalent to k-CNF-Sat modulo serf-reducibility. We propose a complexity class, referred to as Linear Monadic NP, that consists of all problems expressible in existential monadic second order logic whose expressions have a linear measure in terms of a complexity parameter, which is usually the universe size of the problem.
This research direction can be traced back to Fagin's celebrated theorem stating that NP coincides with the class of problems expressible in existential second order logic. Monadic NP, a well-studied class in the literature, is the restriction of the aforementioned logic fragment to existential monadic second order logic. The proposed class Linear Monadic NP is then the restriction of Monadic NP to problems whose expressions have linear measure in the complexity parameter.
We show that Linear Monadic NP includes many natural complete problems such as the satisfiability of linear-size circuits, dominating set, independent dominating set, and perfect code. Therefore, for any of these problems, its subexponential-time solvability is equivalent to the failure of ETH. We prove, using logic games, that the aforementioned problems are inexpressible in the monadic fragment of SNP, and hence, are not captured by the framework of Impagliazzo et al. Finally, we show that Feedback Vertex Set is inexpressible in existential monadic second order logic, and hence is not in Linear Monadic NP, and investigate the existence of certain reductions between Feedback Vertex Set (and variants of it) and 3-CNF-Sat.

Robert Ganian, Ronald de Haan, Iyad Kanj, and Stefan Szeider. On Existential MSO and its Relation to ETH. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 42:1-42:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ganian_et_al:LIPIcs.MFCS.2016.42, author = {Ganian, Robert and de Haan, Ronald and Kanj, Iyad and Szeider, Stefan}, title = {{On Existential MSO and its Relation to ETH}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {42:1--42:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.42}, URN = {urn:nbn:de:0030-drops-64556}, doi = {10.4230/LIPIcs.MFCS.2016.42}, annote = {Keywords: exponential time hypothesis (ETH), monadic second order logic, subexponential time complexity, serf-reducibility, logic games} }

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**Published in:** LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)

Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization.

Eduard Eiben, Robert Ganian, and Stefan Szeider. Meta-kernelization using Well-structured Modulators. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 114-126, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{eiben_et_al:LIPIcs.IPEC.2015.114, author = {Eiben, Eduard and Ganian, Robert and Szeider, Stefan}, title = {{Meta-kernelization using Well-structured Modulators}}, booktitle = {10th International Symposium on Parameterized and Exact Computation (IPEC 2015)}, pages = {114--126}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-92-7}, ISSN = {1868-8969}, year = {2015}, volume = {43}, editor = {Husfeldt, Thore and Kanj, Iyad}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.114}, URN = {urn:nbn:de:0030-drops-55769}, doi = {10.4230/LIPIcs.IPEC.2015.114}, annote = {Keywords: Kernelization, Parameterized complexity, Structural parameters, Rank-width, Split decompositions} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

The modular treewidth of a graph is its treewidth after the contraction of modules. Modular treewidth properly generalizes treewidth and is itself properly generalized by clique-width. We show that the number of satisfying assignments of a CNF formula whose incidence graph has bounded modular treewidth can be computed in polynomial time. This provides new tractable classes of formulas for which #SAT is polynomial. In particular, our result generalizes known results for the treewidth of incidence graphs and is incomparable with known results for clique-width (or rank-width) of signed incidence graphs. The contraction of modules is an effective data reduction procedure. Our algorithm is the first one to harness this technique for #SAT. The order of the polynomial time bound of our algorithm depends on the modular treewidth. We show that this dependency cannot be avoided subject to an assumption from Parameterized Complexity.

Daniel Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model Counting for CNF Formulas of Bounded Modular Treewidth. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 55-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{paulusma_et_al:LIPIcs.STACS.2013.55, author = {Paulusma, Daniel and Slivovsky, Friedrich and Szeider, Stefan}, title = {{Model Counting for CNF Formulas of Bounded Modular Treewidth}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {55--66}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.55}, URN = {urn:nbn:de:0030-drops-39226}, doi = {10.4230/LIPIcs.STACS.2013.55}, annote = {Keywords: Satisfiability, Model Counting, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

The class q-Horn, introduced by Boros, Crama and Hammer in 1990, is one of the largest known classes of propositional CNF formulas for which satisfiability can be decided in polynomial time. This class properly contains the fundamental classes of Horn and Krom formulas as well as the class of renamable (or disguised) Horn formulas.
In this paper we extend this class so that its favorable algorithmic properties can be made accessible to formulas that are outside but "close"' to this class. We show that deciding satisfiability is fixed-parameter tractable parameterized by the distance of the given formula from q-Horn. The distance is measured by the smallest number of variables that we need to delete from the formula in order to get a q-Horn formula, i.e., the size of a smallest deletion backdoor set into the class q-Horn.
This result generalizes known fixed-parameter tractability results for satisfiability decision with respect to the parameters distance from Horn, Krom, and renamable Horn.

Serge Gaspers, Sebastian Ordyniak, M. S. Ramanujan, Saket Saurabh, and Stefan Szeider. Backdoors to q-Horn. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 67-79, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{gaspers_et_al:LIPIcs.STACS.2013.67, author = {Gaspers, Serge and Ordyniak, Sebastian and Ramanujan, M. S. and Saurabh, Saket and Szeider, Stefan}, title = {{Backdoors to q-Horn}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {67--79}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.67}, URN = {urn:nbn:de:0030-drops-39236}, doi = {10.4230/LIPIcs.STACS.2013.67}, annote = {Keywords: Algorithms and data structures, Backdoor sets, Satisfiability, Fixed Parameter Tractability} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with so-called ``beta-acyclic hypergraphs'' can be decided in polynomial time.
We also study the parameterized complexity of SAT for ``almost'' beta-acyclic instances, using as parameter the formula's distance from being beta-acyclic. As distance we use the size of smallest strong backdoor sets and the beta-hypertree width. As a by-product we obtain the W[1]-hardness of SAT parameterized by the (undirected) clique-width of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve (Discr. Appl. Math. 156, 2008).

Sebastian Ordyniak, Daniel Paulusma, and Stefan Szeider. Satisfiability of Acyclic and Almost Acyclic CNF Formulas. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 84-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{ordyniak_et_al:LIPIcs.FSTTCS.2010.84, author = {Ordyniak, Sebastian and Paulusma, Daniel and Szeider, Stefan}, title = {{Satisfiability of Acyclic and Almost Acyclic CNF Formulas}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {84--95}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.84}, URN = {urn:nbn:de:0030-drops-28556}, doi = {10.4230/LIPIcs.FSTTCS.2010.84}, annote = {Keywords: Satisfiability, chordal bipartite graphs, beta-acyclic hypergraphs, backdoor sets, parameterized complexity} }

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