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**Published in:** LIPIcs, Volume 312, 24th International Workshop on Algorithms in Bioinformatics (WABI 2024)

Inverse folding is a classic instance of negative RNA design which consists in finding a sequence that uniquely folds into a target secondary structure with respect to energy minimization. A breakthrough result of Bonnet et al. shows that, even in simple base pairs-based (BP) models, the decision version of a mildly constrained version of inverse folding is NP-hard.
In this work, we show that inverse folding can be solved in linear time for a large collection of targets, including every structure that contains no isolated BP and no isolated stack (or, equivalently, when all helices consist of 3^{+} base pairs). For structures featuring shorter helices, our linear algorithm is no longer guaranteed to produce a solution, but still does so for a large proportion of instances.
Our approach introduces a notion of modulo m-separability, generalizing a property pioneered by Hales et al. Separability is a sufficient condition for the existence of a solution to the inverse folding problem. We show that, for any input secondary structure of length n, a modulo m-separated sequence can be produced in time 𝒪(n 2^m) anytime such a sequence exists. Meanwhile, we show that any structure consisting of 3^{+} base pairs is either trivially non-designable, or always admits a modulo-2 separated solution (m = 2). Solution sequences can thus be produced in linear time, and even be uniformly generated within the set of modulo-2 separable sequences.

Théo Boury, Laurent Bulteau, and Yann Ponty. RNA Inverse Folding Can Be Solved in Linear Time for Structures Without Isolated Stacks or Base Pairs. In 24th International Workshop on Algorithms in Bioinformatics (WABI 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 312, pp. 19:1-19:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{boury_et_al:LIPIcs.WABI.2024.19, author = {Boury, Th\'{e}o and Bulteau, Laurent and Ponty, Yann}, title = {{RNA Inverse Folding Can Be Solved in Linear Time for Structures Without Isolated Stacks or Base Pairs}}, booktitle = {24th International Workshop on Algorithms in Bioinformatics (WABI 2024)}, pages = {19:1--19:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-340-9}, ISSN = {1868-8969}, year = {2024}, volume = {312}, editor = {Pissis, Solon P. and Sung, Wing-Kin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2024.19}, URN = {urn:nbn:de:0030-drops-206632}, doi = {10.4230/LIPIcs.WABI.2024.19}, annote = {Keywords: RNA structure, String Design, Parameterized Complexity, Uniform Sampling} }

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Complete Volume

**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

LIPIcs, Volume 259, CPM 2023, Complete Volume

34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 1-472, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Proceedings{bulteau_et_al:LIPIcs.CPM.2023, title = {{LIPIcs, Volume 259, CPM 2023, Complete Volume}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {1--472}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023}, URN = {urn:nbn:de:0030-drops-179536}, doi = {10.4230/LIPIcs.CPM.2023}, annote = {Keywords: LIPIcs, Volume 259, CPM 2023, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 259, 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)

Front Matter, Table of Contents, Preface, Conference Organization

34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2023.0, author = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {0:i--0:xvi}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-276-1}, ISSN = {1868-8969}, year = {2023}, volume = {259}, editor = {Bulteau, Laurent and Lipt\'{a}k, Zsuzsanna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2023.0}, URN = {urn:nbn:de:0030-drops-179542}, doi = {10.4230/LIPIcs.CPM.2023.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 242, 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)

Despite being a textbook application of dynamic programming (DP) and routine task in RNA structure analysis, RNA secondary structure prediction remains challenging whenever pseudoknots come into play. To circumvent the NP-hardness of energy minimization in realistic energy models, specialized algorithms have been proposed for restricted conformation classes that capture the most frequently observed configurations.
While these methods rely on hand-crafted DP schemes, we generalize and fully automatize the design of DP pseudoknot prediction algorithms. We formalize the problem of designing DP algorithms for an (infinite) class of conformations, modeled by (a finite number of) fatgraphs, and automatically build DP schemes minimizing their algorithmic complexity. We propose an algorithm for the problem, based on the tree-decomposition of a well-chosen representative structure, which we simplify and reinterpret as a DP scheme. The algorithm is fixed-parameter tractable for the tree-width tw of the fatgraph, and its output represents a 𝒪(n^{tw+1}) algorithm for predicting the MFE folding of an RNA of length n.
Our general framework supports general energy models, partition function computations, recursive substructures and partial folding, and could pave the way for algebraic dynamic programming beyond the context-free case.

Bertrand Marchand, Sebastian Will, Sarah J. Berkemer, Laurent Bulteau, and Yann Ponty. Automated Design of Dynamic Programming Schemes for RNA Folding with Pseudoknots. In 22nd International Workshop on Algorithms in Bioinformatics (WABI 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 242, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{marchand_et_al:LIPIcs.WABI.2022.7, author = {Marchand, Bertrand and Will, Sebastian and Berkemer, Sarah J. and Bulteau, Laurent and Ponty, Yann}, title = {{Automated Design of Dynamic Programming Schemes for RNA Folding with Pseudoknots}}, booktitle = {22nd International Workshop on Algorithms in Bioinformatics (WABI 2022)}, pages = {7:1--7:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-243-3}, ISSN = {1868-8969}, year = {2022}, volume = {242}, editor = {Boucher, Christina and Rahmann, Sven}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2022.7}, URN = {urn:nbn:de:0030-drops-170414}, doi = {10.4230/LIPIcs.WABI.2022.7}, annote = {Keywords: RNA folding, treewidth, dynamic programming} }

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**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

In the NP-hard Longest Common Subsequence problem (LCS), given a set of strings, the task is to find a string that can be obtained from every input string using as few deletions as possible. LCS is one of the most fundamental string problems with numerous applications in various areas, having gained a lot of attention in the algorithms and complexity research community. Significantly improving on an algorithm by Irving and Fraser [CPM'92], featured as a research challenge in a 2014 survey paper, we show that LCS is fixed-parameter tractable (FPT) when parameterized by the maximum number of deletions per input string. Given the relatively moderate running time of our algorithm (linear time when the parameter is a constant) and small parameter values to be expected in several applications, we believe that our purely theoretical analysis could finally pave the way to a new, exact and practically useful algorithm for this notoriously hard string problem.

Laurent Bulteau, Mark Jones, Rolf Niedermeier, and Till Tantau. An FPT-Algorithm for Longest Common Subsequence Parameterized by the Maximum Number of Deletions. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 6:1-6:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2022.6, author = {Bulteau, Laurent and Jones, Mark and Niedermeier, Rolf and Tantau, Till}, title = {{An FPT-Algorithm for Longest Common Subsequence Parameterized by the Maximum Number of Deletions}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {6:1--6:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.6}, URN = {urn:nbn:de:0030-drops-161338}, doi = {10.4230/LIPIcs.CPM.2022.6}, annote = {Keywords: NP-hard string problems, multiple sequence alignment, center string, parameterized complexity, search tree algorithms, enumerative algorithms} }

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**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

We study in this paper the Doubly Partially Ordered Pattern Matching (or DPOP Matching) problem, a natural extension of the Permutation Pattern Matching problem. Permutation Pattern Matching takes as input two permutations σ and π, and asks whether there exists an occurrence of σ in π; whereas DPOP Matching takes two partial orders P_v and P_p defined on the same set X and a permutation π, and asks whether there exist |X| elements in π whose values (resp., positions) are in accordance with P_v (resp., P_p). Posets P_v and P_p aim at relaxing the conditions formerly imposed by the permutation σ, since σ yields a total order on both positions and values. Our problem being NP-hard in general (as Permutation Pattern Matching is), we consider restrictions on several parameters/properties of the input, e.g., bounding the size of the pattern, assuming symmetry of the posets (i.e., P_v and P_p are identical), assuming that one partial order is a total (resp., weak) order, bounding the length of the longest chain/anti-chain in the posets, or forbidding specific patterns in π. For each such restriction, we provide results which together give a(n almost) complete landscape for the algorithmic complexity of the problem.

Laurent Bulteau, Guillaume Fertin, Vincent Jugé, and Stéphane Vialette. Permutation Pattern Matching for Doubly Partially Ordered Patterns. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2022.21, author = {Bulteau, Laurent and Fertin, Guillaume and Jug\'{e}, Vincent and Vialette, St\'{e}phane}, title = {{Permutation Pattern Matching for Doubly Partially Ordered Patterns}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {21:1--21:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.21}, URN = {urn:nbn:de:0030-drops-161481}, doi = {10.4230/LIPIcs.CPM.2022.21}, annote = {Keywords: Partial orders, Permutations, Pattern Matching, Algorithmic Complexity, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

In this article, we study two problems consisting in reordering a tree to fit with an order on its leaves provided as input, which were earlier introduced in the context of phylogenetic tree comparison for bioinformatics, OTCM and OTDE. The first problem consists in finding an order which minimizes the number of inversions with an input order on the leaves, while the second one consists in removing the minimum number of leaves from the tree to make it consistent with the input order on the remaining leaves. We show that both problems are NP-complete when the maximum degree is not bounded, as well as a problem on tree alignment, answering two questions opened in 2010 by Henning Fernau, Michael Kaufmann and Mathias Poths. We provide a polynomial-time algorithm for OTDE in the case where the maximum degree is bounded by a constant and an FPT algorithm in a parameter lower than the number of leaves to delete. Our results have practical interest not only for bioinformatics but also for digital humanities to evaluate, for example, the consistency of the dendrogram obtained from a hierarchical clustering algorithm with a chronological ordering of its leaves. We explore the possibilities of practical use of our results both on trees obtained by clustering the literary works of French authors and on simulated data, using implementations of our algorithms in Python.

Laurent Bulteau, Philippe Gambette, and Olga Seminck. Reordering a Tree According to an Order on Its Leaves. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2022.24, author = {Bulteau, Laurent and Gambette, Philippe and Seminck, Olga}, title = {{Reordering a Tree According to an Order on Its Leaves}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.24}, URN = {urn:nbn:de:0030-drops-161516}, doi = {10.4230/LIPIcs.CPM.2022.24}, annote = {Keywords: tree, clustering, order, permutation, inversions, FPT algorithm, NP-hardness, tree drawing, OTCM, OTDE, TTDE} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

In this paper, we study the Independent Set (IS) reconfiguration problem in graphs. An IS reconfiguration is a scenario transforming an IS L into another IS R, inserting/removing vertices one step at a time while keeping the cardinalities of intermediate sets greater than a specified threshold. We focus on the bipartite variant where only start and end vertices are allowed in intermediate ISs. Our motivation is an application to the RNA energy barrier problem from bioinformatics, for which a natural parameter would be the difference between the initial IS size and the threshold.
We first show the para-NP hardness of the problem with respect to this parameter. We then investigate a new parameter, the cardinality range, denoted by ρ which captures the maximum deviation of the reconfiguration scenario from optimal sets (formally, ρ is the maximum difference between the cardinalities of an intermediate IS and an optimal IS). We give two different routes to show that this problem is in XP for ρ: The first is a direct O(n²)-space, O(n^{2ρ+2.5})-time algorithm based on a separation lemma; The second builds on a parameterized equivalence with the directed pathwidth problem, leading to a O(n^{ρ+1})-space, O(n^{ρ+2})-time algorithm for the reconfiguration problem through an adaptation of a prior result by Tamaki [Tamaki, 2011]. This equivalence is an interesting result in its own right, connecting a reconfiguration problem (which is essentially a connectivity problem within a reconfiguration network) with a structural parameter for an auxiliary graph.
We demonstrate the practicality of these algorithms, and the relevance of our introduced parameter, by considering the application of our algorithms on random small-degree instances for our problem. Moreover, we reformulate the computation of the energy barrier between two RNA secondary structures, a classic hard problem in computational biology, as an instance of bipartite reconfiguration. Our results on IS reconfiguration thus yield an XP algorithm in O(n^{ρ+2}) for the energy barrier problem, improving upon a partial O(n^{2ρ+2.5}) algorithm for the problem.

Laurent Bulteau, Bertrand Marchand, and Yann Ponty. A New Parametrization for Independent Set Reconfiguration and Applications to RNA Kinetics. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bulteau_et_al:LIPIcs.IPEC.2021.11, author = {Bulteau, Laurent and Marchand, Bertrand and Ponty, Yann}, title = {{A New Parametrization for Independent Set Reconfiguration and Applications to RNA Kinetics}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {11:1--11:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.11}, URN = {urn:nbn:de:0030-drops-153946}, doi = {10.4230/LIPIcs.IPEC.2021.11}, annote = {Keywords: reconfiguration problems - parameterized algorithms - RNA bioinformatics - directed pathwidth} }

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**Published in:** LIPIcs, Volume 201, 21st International Workshop on Algorithms in Bioinformatics (WABI 2021)

Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed-Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the Tree-Diet problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth tw'. Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account.
Our core result is an FPT dynamic programming algorithm for Tree-Diet, using 2^{O(tw)}n time and space. We complement this result with parameterized complexity lower-bounds for stronger variants (e.g., NP-hardness when tw' or tw-tw' is constant). We propose a prototype implementation for our approach which we apply on difficult instances of selected RNA-based problems: RNA design, sequence-structure alignment, and search of pseudoknotted RNAs in genomes, revealing very encouraging results. This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics.

Bertrand Marchand, Yann Ponty, and Laurent Bulteau. Tree Diet: Reducing the Treewidth to Unlock FPT Algorithms in RNA Bioinformatics. In 21st International Workshop on Algorithms in Bioinformatics (WABI 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 201, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{marchand_et_al:LIPIcs.WABI.2021.7, author = {Marchand, Bertrand and Ponty, Yann and Bulteau, Laurent}, title = {{Tree Diet: Reducing the Treewidth to Unlock FPT Algorithms in RNA Bioinformatics}}, booktitle = {21st International Workshop on Algorithms in Bioinformatics (WABI 2021)}, pages = {7:1--7:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-200-6}, ISSN = {1868-8969}, year = {2021}, volume = {201}, editor = {Carbone, Alessandra and El-Kebir, Mohammed}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2021.7}, URN = {urn:nbn:de:0030-drops-143604}, doi = {10.4230/LIPIcs.WABI.2021.7}, annote = {Keywords: RNA, treewidth, FPT algorithms, RNA design, structure-sequence alignment} }

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**Published in:** LIPIcs, Volume 191, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)

The additive x-disorder of a permutation is the sum of the absolute differences of all pairs of consecutive elements. We show that the additive x-disorder of a permutation of S(n), n ≥ 2, ranges from n-1 to ⌊n²/2⌋ - 1, and we give a complete characterization of permutations having extreme such values. Moreover, for any positive integers n and d such that n ≥ 2 and n-1 ≤ d ≤ ⌊n²/2⌋ - 1, we propose a linear-time algorithm to compute a permutation π ∈ S(n) with additive x-disorder d.

Laurent Bulteau, Samuele Giraudo, and Stéphane Vialette. Disorders and Permutations. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 11:1-11:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2021.11, author = {Bulteau, Laurent and Giraudo, Samuele and Vialette, St\'{e}phane}, title = {{Disorders and Permutations}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {11:1--11:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-186-3}, ISSN = {1868-8969}, year = {2021}, volume = {191}, editor = {Gawrychowski, Pawe{\l} and Starikovskaya, Tatiana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2021.11}, URN = {urn:nbn:de:0030-drops-139628}, doi = {10.4230/LIPIcs.CPM.2021.11}, annote = {Keywords: Permutation, Algorithm} }

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**Published in:** LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

A partition (V_1,...,V_k) of the vertex set of a graph G with a (not necessarily proper) colouring c is colourful if no two vertices in any V_i have the same colour and every set V_i induces a connected graph. The Colourful Partition problem, introduced by Adamaszek and Popa, is to decide whether a coloured graph (G,c) has a colourful partition of size at most k. This problem is related to the Colourful Components problem, introduced by He, Liu and Zhao, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most p edges.
Despite the similarities in their definitions, we show that Colourful Partition and Colourful Components may have different complexities for restricted instances. We tighten known NP-hardness results for both problems by closing a number of complexity gaps. In addition, we prove new hardness and tractability results for Colourful Partition. In particular, we prove that deciding whether a coloured graph (G,c) has a colourful partition of size 2 is NP-complete for coloured planar bipartite graphs of maximum degree 3 and path-width 3, but polynomial-time solvable for coloured graphs of treewidth 2.
Rather than performing an ad hoc study, we use our classical complexity results to guide us in undertaking a thorough parameterized study of Colourful Partition. We show that this leads to suitable parameters for obtaining FPT results and moreover prove that Colourful Components and Colourful Partition may have different parameterized complexities, depending on the chosen parameter.

Laurent Bulteau, Konrad K. Dabrowski, Guillaume Fertin, Matthew Johnson, Daniël Paulusma, and Stéphane Vialette. Finding a Small Number of Colourful Components. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2019.20, author = {Bulteau, Laurent and Dabrowski, Konrad K. and Fertin, Guillaume and Johnson, Matthew and Paulusma, Dani\"{e}l and Vialette, St\'{e}phane}, title = {{Finding a Small Number of Colourful Components}}, booktitle = {30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)}, pages = {20:1--20:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-103-0}, ISSN = {1868-8969}, year = {2019}, volume = {128}, editor = {Pisanti, Nadia and P. Pissis, Solon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.20}, URN = {urn:nbn:de:0030-drops-104914}, doi = {10.4230/LIPIcs.CPM.2019.20}, annote = {Keywords: Colourful component, colourful partition, tree, treewidth, vertex cover} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The parameterised complexity of consensus string problems (Closest String, Closest Substring, Closest String with Outliers) is investigated in a more general setting, i. e., with a bound on the maximum Hamming distance and a bound on the sum of Hamming distances between solution and input strings. We completely settle the parameterised complexity of these generalised variants of Closest String and Closest Substring, and partly for Closest String with Outliers; in addition, we answer some open questions from the literature regarding the classical problem variants with only one distance bound. Finally, we investigate the question of polynomial kernels and respective lower bounds.

Laurent Bulteau and Markus L. Schmid. Consensus Strings with Small Maximum Distance and Small Distance Sum. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bulteau_et_al:LIPIcs.MFCS.2018.1, author = {Bulteau, Laurent and Schmid, Markus L.}, title = {{Consensus Strings with Small Maximum Distance and Small Distance Sum}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {1:1--1:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.1}, URN = {urn:nbn:de:0030-drops-95834}, doi = {10.4230/LIPIcs.MFCS.2018.1}, annote = {Keywords: Consensus String Problems, Closest String, Closest Substring, Parameterised Complexity, Kernelisation} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

In Genomic Scaffold Filling, one aims at polishing in silico a draft genome, called scaffold. The scaffold is given in the form of an ordered set of gene sequences, called contigs. This is done by confronting the scaffold to an already complete reference genome from a close species. More precisely, given a scaffold S, a reference genome G and a score function f() between two genomes, the aim is to complete S by adding the missing genes from G so that the obtained complete genome S* optimizes f(S*, G). In this paper, we extend a model of Jiang et al. [CPM 2016] (i) by allowing the insertions of strings instead of single characters (i.e., some groups of genes may be forced to be inserted together) and (ii) by considering two alternative score functions: the first generalizes the notion of common adjacencies by maximizing the number of common k-mers between S* and G (k-Mer Scaffold Filling), the second aims at minimizing the number of breakpoints between S* and G (Min-Breakpoint Scaffold Filling). We study these problems from the parameterized complexity point of view, providing fixed-parameter (FPT) algorithms for both problems. In particular, we show that k-Mer Scaffold Filling is FPT wrt. parameter l, the number of additional k-mers realized by the completion of S—this answers an open question of Jiang et al. [CPM 2016]. We also show that Min-Breakpoint Scaffold Filling is FPT wrt. a parameter combining the number of missing genes, the number of gene repetitions and the target distance.

Laurent Bulteau, Guillaume Fertin, and Christian Komusiewicz. Beyond Adjacency Maximization: Scaffold Filling for New String Distances. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bulteau_et_al:LIPIcs.CPM.2017.27, author = {Bulteau, Laurent and Fertin, Guillaume and Komusiewicz, Christian}, title = {{Beyond Adjacency Maximization: Scaffold Filling for New String Distances}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {27:1--27:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.27}, URN = {urn:nbn:de:0030-drops-73364}, doi = {10.4230/LIPIcs.CPM.2017.27}, annote = {Keywords: computational biology, strings, FPT algorithms, kernelization} }

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