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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

PPSZ is the fastest known algorithm for (d,k)-CSP problems, for most values of d and k. It goes through the variables in random order and sets each variable randomly to one of the d colors, excluding those colors that can be ruled out by looking at few constraints at a time.
We propose and analyze a modification of PPSZ: whenever all but 2 colors can be ruled out for some variable, immediately set that variable randomly to one of the remaining colors. We show that our new "impatient PPSZ" outperforms PPSZ exponentially for all k and all d ≥ 3 on formulas with a unique satisfying assignment.

Shibo Li and Dominik Scheder. Impatient PPSZ - A Faster Algorithm for CSP. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{li_et_al:LIPIcs.ISAAC.2021.33, author = {Li, Shibo and Scheder, Dominik}, title = {{Impatient PPSZ - A Faster Algorithm for CSP}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {33:1--33:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.33}, URN = {urn:nbn:de:0030-drops-154662}, doi = {10.4230/LIPIcs.ISAAC.2021.33}, annote = {Keywords: Randomized algorithms, Constraint Satisfaction Problems, exponential-time algorithms} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We construct k-CNFs with m variables on which the strong version of PPSZ k-SAT algorithm, which uses resolution of width bounded by O(√{log log m}), has success probability at most 2^{-(1-(1 + ε)2/k)m} for every ε > 0. Previously such a bound was known only for the weak PPSZ algorithm which exhaustively searches through small subformulas of the CNF to see if any of them forces the value of a given variable, and for strong PPSZ the best known previous upper bound was 2^{-(1-O(log(k)/k))m} (Pudlák et al., ICALP 2017).

Dominik Scheder and Navid Talebanfard. Super Strong ETH Is True for PPSZ with Small Resolution Width. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 3:1-3:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{scheder_et_al:LIPIcs.CCC.2020.3, author = {Scheder, Dominik and Talebanfard, Navid}, title = {{Super Strong ETH Is True for PPSZ with Small Resolution Width}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {3:1--3:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.3}, URN = {urn:nbn:de:0030-drops-125552}, doi = {10.4230/LIPIcs.CCC.2020.3}, annote = {Keywords: k-SAT, PPSZ, Resolution} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Cryptogenography is a secret-leaking game in which one of n players is holding a secret to be leaked. The n players engage in communication as to (1) reveal the secret while (2) keeping the identity of the secret holder as obscure as possible. All communication is public, and no computational hardness assumptions are made, i.e., the setting is purely information theoretic. Brody, Jakobsen, Scheder, and Winkler [Joshua Brody et al., 2014] formally defined this problem, showed that it has an equivalent geometric characterization, and gave upper and lower bounds for the case in which the n players want to leak a single bit. Surprisingly, even the easiest case, where two players want to leak a secret consisting of a single bit, is not completely understood. Doerr and Künnemann [Benjamin Doerr and Marvin Künnemann, 2016] showed how to automatically search for good protocols using a computer, thus finding an improved protocol for the 1-bit two-player case. In this work, we show how the search for upper bounds (impossibility results) can be formulated as a Sum of Squares program. We implement this idea for the 1-bit two-player case and significantly improve the previous upper bound from 47/128 = 0.3671875 to 0.35183.

Dominik Scheder, Shuyang Tang, and Jiaheng Zhang. Searching for Cryptogenography Upper Bounds via Sum of Square Programming. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 31:1-31:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{scheder_et_al:LIPIcs.ISAAC.2019.31, author = {Scheder, Dominik and Tang, Shuyang and Zhang, Jiaheng}, title = {{Searching for Cryptogenography Upper Bounds via Sum of Square Programming}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {31:1--31:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.31}, URN = {urn:nbn:de:0030-drops-115276}, doi = {10.4230/LIPIcs.ISAAC.2019.31}, annote = {Keywords: Communication Complexity, Secret Leaking, Sum of Squares Programming} }

Document

**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

The currently fastest known algorithm for k-SAT is PPSZ named after its inventors Paturi, Pudlak, Saks, and Zane. Analyzing its running time is much easier for input formulas with a unique satisfying assignment. In this paper, we achieve three goals. First, we simplify Hertli's analysis for input formulas with multiple satisfying assignments. Second, we show a "translation result": if you improve PPSZ for k-CNF formulas with a unique satisfying assignment, you will immediately get a (weaker) improvement for general k-CNF formulas. Combining this with a result by Hertli from 2014, in which he gives an algorithm for Unique-3-SAT slightly beating PPSZ, we obtain an algorithm beating PPSZ for general 3-SAT, thus obtaining the so far best known worst-case bounds for 3-SAT.

Dominik Scheder and John P. Steinberger. PPSZ for General k-SAT - Making Hertli's Analysis Simpler and 3-SAT Faster. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{scheder_et_al:LIPIcs.CCC.2017.9, author = {Scheder, Dominik and Steinberger, John P.}, title = {{PPSZ for General k-SAT - Making Hertli's Analysis Simpler and 3-SAT Faster}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {9:1--9:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.9}, URN = {urn:nbn:de:0030-drops-75355}, doi = {10.4230/LIPIcs.CCC.2017.9}, annote = {Keywords: Boolean satisfiability, exponential algorithms, randomized algorithms} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We construct uniquely satisfiable k-CNF formulas that are hard for the PPSZ algorithm, the currently best known algorithm solving k-SAT. This algorithm tries to generate a satisfying assignment by picking a random variable at a time and attempting to derive its value using some inference heuristic and otherwise assigning a random value. The "weak PPSZ" checks all subformulas of a given size to derive a value and the "strong PPSZ" runs resolution with width bounded by some given function. Firstly, we construct graph-instances on which "weak PPSZ" has savings of at most (2 + epsilon)/k; the saving of an algorithm on an input formula with n variables is the largest gamma such that the algorithm succeeds (i.e. finds a satisfying assignment) with probability at least 2^{- (1 - gamma) n}. Since PPSZ (both weak and strong) is known to have savings of at least (pi^2 + o(1))/6k, this is optimal up to the constant factor. In particular, for k=3, our upper bound is 2^{0.333... n}, which is fairly close to the lower bound 2^{0.386... n} of Hertli [SIAM J. Comput.'14]. We also construct instances based on linear systems over F_2 for which strong PPSZ has savings of at most O(log(k)/k). This is only a log(k) factor away from the optimal bound. Our constructions improve previous savings upper bound of O((log^2(k))/k) due to Chen et al. [SODA'13].

Pavel Pudlák, Dominik Scheder, and Navid Talebanfard. Tighter Hard Instances for PPSZ. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 85:1-85:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{pudlak_et_al:LIPIcs.ICALP.2017.85, author = {Pudl\'{a}k, Pavel and Scheder, Dominik and Talebanfard, Navid}, title = {{Tighter Hard Instances for PPSZ}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {85:1--85:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.85}, URN = {urn:nbn:de:0030-drops-74144}, doi = {10.4230/LIPIcs.ICALP.2017.85}, annote = {Keywords: k-SAT, Strong Exponential Time Hypothesis, PPSZ, Resolution} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

A critical variable of a satisfiable CNF formula is a variable that has the same value in all satisfying assignments. Using a simple case distinction on the fraction of critical variables of a CNF formula, we improve the running time for 3-SAT from O(1.32216^n) [Rolf 2006] to O(1.32153^n). Using a different approach, Iwama et al. very recently achieved a running time of O(1.32113^n). Our method nicely combines with theirs, yielding the currently fastest known algorithm with running time O(1.32065^n). We also improve the bound for 4-SAT from O(1.47390^n) [Hofmeister et al. 2002] to O(1.46928^n), where O(1.46981^n) can be obtained using the methods of Hofmeister and Rolf.

Timon Hertli, Robin A. Moser, and Dominik Scheder. Improving PPSZ for 3-SAT using Critical Variables. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 237-248, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{hertli_et_al:LIPIcs.STACS.2011.237, author = {Hertli, Timon and Moser, Robin A. and Scheder, Dominik}, title = {{Improving PPSZ for 3-SAT using Critical Variables}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {237--248}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.237}, URN = {urn:nbn:de:0030-drops-30147}, doi = {10.4230/LIPIcs.STACS.2011.237}, annote = {Keywords: SAT, satisfiability, randomized, exponential time, algorithm, 3-SAT, 4-SAT} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

We call a CNF formula {\em linear} if any two clauses have at most one variable in common. We show that there exist unsatisfiable linear $k$-CNF formulas with at most $4k^24^k$ clauses, and on the other hand, any linear $k$-CNF formula with at most $\frac{4^k}{8e^2k^2}$ clauses is satisfiable. The upper bound uses probabilistic means, and we have no explicit construction coming even close to it. One reason for this is that unsatisfiable linear formulas exhibit a more complex structure than general (non-linear) formulas: First, any treelike resolution refutation of any unsatisfiable linear $k$-CNF formula has size at least $2^{2^{\frac{k}{2}-1}}$. This implies that small unsatisfiable linear $k$-CNF formulas are hard instances for Davis-Putnam style splitting algorithms. Second, if we require that the formula $F$ have a {\em strict} resolution tree, i.e. every clause of $F$ is used only once in the resolution tree, then we need at least $a^{a^{\iddots^a}}$ clauses, where $a \approx 2$ and the height of this tower is roughly $k$.

Dominik Scheder. Unsatisfiable Linear CNF Formulas Are Large and Complex. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 621-632, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{scheder:LIPIcs.STACS.2010.2490, author = {Scheder, Dominik}, title = {{Unsatisfiable Linear CNF Formulas Are Large and Complex}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {621--632}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2490}, URN = {urn:nbn:de:0030-drops-24901}, doi = {10.4230/LIPIcs.STACS.2010.2490}, annote = {Keywords: Extremal combinatorics, proof complexity, probabilistic method} }

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