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**Published in:** LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

Given a conjunctive query Q and a database 𝐃, a direct access to the answers of Q over 𝐃 is the operation of returning, given an index j, the j-th answer for some order on its answers. While this problem is #P-hard in general with respect to combined complexity, many conjunctive queries have an underlying structure that allows for a direct access to their answers for some lexicographical ordering that takes polylogarithmic time in the size of the database after a polynomial time precomputation. Previous work has precisely characterised the tractable classes and given fine-grained lower bounds on the precomputation time needed depending on the structure of the query. In this paper, we generalise these tractability results to the case of signed conjunctive queries, that is, conjunctive queries that may contain negative atoms. Our technique is based on a class of circuits that can represent relational data. We first show that this class supports tractable direct access after a polynomial time preprocessing. We then give bounds on the size of the circuit needed to represent the answer set of signed conjunctive queries depending on their structure. Both results combined together allow us to prove the tractability of direct access for a large class of conjunctive queries. On the one hand, we recover the known tractable classes from the literature in the case of positive conjunctive queries. On the other hand, we generalise and unify known tractability results about negative conjunctive queries - that is, queries having only negated atoms. In particular, we show that the class of β-acyclic negative conjunctive queries and the class of bounded nest set width negative conjunctive queries admit tractable direct access.

Florent Capelli and Oliver Irwin. Direct Access for Conjunctive Queries with Negations. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{capelli_et_al:LIPIcs.ICDT.2024.13, author = {Capelli, Florent and Irwin, Oliver}, title = {{Direct Access for Conjunctive Queries with Negations}}, booktitle = {27th International Conference on Database Theory (ICDT 2024)}, pages = {13:1--13:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-312-6}, ISSN = {1868-8969}, year = {2024}, volume = {290}, editor = {Cormode, Graham and Shekelyan, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.13}, URN = {urn:nbn:de:0030-drops-197958}, doi = {10.4230/LIPIcs.ICDT.2024.13}, annote = {Keywords: Conjunctive queries, factorized databases, direct access, hypertree decomposition} }

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**Published in:** LIPIcs, Volume 290, 27th International Conference on Database Theory (ICDT 2024)

We study the problem of enumerating the satisfying assignments for certain circuit classes from knowledge compilation, where assignments are ranked in a specific order. In particular, we show how this problem can be used to efficiently perform ranked enumeration of the answers to MSO queries over trees, with the order being given by a ranking function satisfying a subset-monotonicity property.
Assuming that the number of variables is constant, we show that we can enumerate the satisfying assignments in ranked order for so-called multivalued circuits that are smooth, decomposable, and in negation normal form (smooth multivalued DNNF). There is no preprocessing and the enumeration delay is linear in the size of the circuit times the number of values, plus a logarithmic term in the number of assignments produced so far. If we further assume that the circuit is deterministic (smooth multivalued d-DNNF), we can achieve linear-time preprocessing in the circuit, and the delay only features the logarithmic term.

Antoine Amarilli, Pierre Bourhis, Florent Capelli, and Mikaël Monet. Ranked Enumeration for MSO on Trees via Knowledge Compilation. In 27th International Conference on Database Theory (ICDT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 290, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{amarilli_et_al:LIPIcs.ICDT.2024.25, author = {Amarilli, Antoine and Bourhis, Pierre and Capelli, Florent and Monet, Mika\"{e}l}, title = {{Ranked Enumeration for MSO on Trees via Knowledge Compilation}}, booktitle = {27th International Conference on Database Theory (ICDT 2024)}, pages = {25:1--25:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-312-6}, ISSN = {1868-8969}, year = {2024}, volume = {290}, editor = {Cormode, Graham and Shekelyan, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2024.25}, URN = {urn:nbn:de:0030-drops-198079}, doi = {10.4230/LIPIcs.ICDT.2024.25}, annote = {Keywords: Enumeration, knowledge compilation, monadic second-order logic} }

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**Published in:** LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

In this paper, we introduce a technique we call geometric amortization for enumeration algorithms, which can be used to make the delay of enumeration algorithms more regular with little overhead on the space it uses. More precisely, we consider enumeration algorithms having incremental linear delay, that is, algorithms enumerating, on input x, a set A(x) such that for every t ≤ ♯ A(x), it outputs at least t solutions in time O(t⋅p(|x|)), where p is a polynomial. We call p the incremental delay of the algorithm. While it is folklore that one can transform such an algorithm into an algorithm with maximal delay O(p(|x|)), the naive transformation may use exponential space. We show that, using geometric amortization, such an algorithm can be transformed into an algorithm with delay O(p(|x|)log(♯A(x))) and space O(s log(♯A(x))) where s is the space used by the original algorithm. In terms of complexity, we prove that classes DelayP and IncP₁ with polynomial space coincide.
We apply geometric amortization to show that one can trade the delay of flashlight search algorithms for their average delay up to a factor of O(log(♯A(x))). We illustrate how this tradeoff is advantageous for the enumeration of solutions of DNF formulas.

Florent Capelli and Yann Strozecki. Geometric Amortization of Enumeration Algorithms. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{capelli_et_al:LIPIcs.STACS.2023.18, author = {Capelli, Florent and Strozecki, Yann}, title = {{Geometric Amortization of Enumeration Algorithms}}, booktitle = {40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)}, pages = {18:1--18:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-266-2}, ISSN = {1868-8969}, year = {2023}, volume = {254}, editor = {Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.18}, URN = {urn:nbn:de:0030-drops-176703}, doi = {10.4230/LIPIcs.STACS.2023.18}, annote = {Keywords: Enumeration, Polynomial Delay, Incremental Delay, Amortization} }

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**Published in:** LIPIcs, Volume 220, 25th International Conference on Database Theory (ICDT 2022)

In this paper, we study the problem of optimizing a linear program whose variables are the answers to a conjunctive query. For this we propose the language LP(CQ) for specifying linear programs whose constraints and objective functions depend on the answer sets of conjunctive queries. We contribute an efficient algorithm for solving programs in a fragment of LP(CQ). The naive approach constructs a linear program having as many variables as there are elements in the answer set of the queries. Our approach constructs a linear program having the same optimal value but fewer variables. This is done by exploiting the structure of the conjunctive queries using generalized hypertree decompositions of small width to factorize elements of the answer set together. We illustrate the various applications of LP(CQ) programs on three examples: optimizing deliveries of resources, minimizing noise for differential privacy, and computing the s-measure of patterns in graphs as needed for data mining.

Florent Capelli, Nicolas Crosetti, Joachim Niehren, and Jan Ramon. Linear Programs with Conjunctive Queries. In 25th International Conference on Database Theory (ICDT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 220, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{capelli_et_al:LIPIcs.ICDT.2022.5, author = {Capelli, Florent and Crosetti, Nicolas and Niehren, Joachim and Ramon, Jan}, title = {{Linear Programs with Conjunctive Queries}}, booktitle = {25th International Conference on Database Theory (ICDT 2022)}, pages = {5:1--5:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-223-5}, ISSN = {1868-8969}, year = {2022}, volume = {220}, editor = {Olteanu, Dan and Vortmeier, Nils}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2022.5}, URN = {urn:nbn:de:0030-drops-158796}, doi = {10.4230/LIPIcs.ICDT.2022.5}, annote = {Keywords: Database queries, linear programming, hypergraph decomposition} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We generalize several tractability results concerning the tractability of Quantified Boolean Formulas (QBF) with restricted underlying structure. To this end, we introduce a notion of width for structured DNNF which are a class of Boolean circuits heavily studied in knowledge compilation, a subarea of artificial intelligence. We then show that structured DNNF allow quantifier elimination with a size blow-up depending only on the width of the DNNF and not its size. Using known algorithms transforming restricted CNF-formulas into deterministic DNNF, we apply this result to generalize several results for counting and decision on QBF. We also complement these results with lower bounds that show that our definitions and results are essentially optimal in several senses.

Florent Capelli and Stefan Mengel. Tractable QBF by Knowledge Compilation. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{capelli_et_al:LIPIcs.STACS.2019.18, author = {Capelli, Florent and Mengel, Stefan}, title = {{Tractable QBF by Knowledge Compilation}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.18}, URN = {urn:nbn:de:0030-drops-102571}, doi = {10.4230/LIPIcs.STACS.2019.18}, annote = {Keywords: QBF, knowledge compilation, parameterized algorithms} }

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**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We show that SAT on beta-acyclic CNF-formulas can be solved in polynomial time. In contrast to previous algorithms for other structurally restricted classes of formulas, our algorithm does not proceed by dynamic programming. Instead, it works along an elimination order, solving a weighted version of constraint satisfaction. We give evidence that this deviation from more standard algorithms is no coincidence by showing that it is outside of the framework recently proposed by Saether et al. (SAT 2014) which subsumes all other structural tractability results for #SAT known so far.

Johann Brault-Baron, Florent Capelli, and Stefan Mengel. Understanding Model Counting for beta-acyclic CNF-formulas. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 143-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{braultbaron_et_al:LIPIcs.STACS.2015.143, author = {Brault-Baron, Johann and Capelli, Florent and Mengel, Stefan}, title = {{Understanding Model Counting for beta-acyclic CNF-formulas}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {143--156}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.143}, URN = {urn:nbn:de:0030-drops-49106}, doi = {10.4230/LIPIcs.STACS.2015.143}, annote = {Keywords: model counting, hypergraph acyclicity, structural tractability} }

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

We investigate the algebraic complexity of tensor calulus. We consider a generalization of iterated matrix product to tensors and show that the resulting formulas exactly capture VP, the class of polynomial families efficiently computable by arithmetic circuits. This gives a natural and robust characterization of this complexity class that despite its naturalness is not very well understood so far.

Florent Capelli, Arnaud Durand, and Stefan Mengel. The arithmetic complexity of tensor contractions. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 365-376, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{capelli_et_al:LIPIcs.STACS.2013.365, author = {Capelli, Florent and Durand, Arnaud and Mengel, Stefan}, title = {{The arithmetic complexity of tensor contractions}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {365--376}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.365}, URN = {urn:nbn:de:0030-drops-39481}, doi = {10.4230/LIPIcs.STACS.2013.365}, annote = {Keywords: algebraic complexity, arithmetic circuits, tensor calculus} }

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