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**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

We analyze the complexity of Arborescence Reconfiguration on temporal digraphs (Temporal Arborescence Reconfiguration). The problem, given two temporal arborescences in a temporal digraph, asks for the minimum number of arc flips, i.e. arc exchanges, that result in a sequence of temporal arborescences that transforms one into the other. We analyze the complexity of the problem, taking into account also its approximation and parameterized complexity, even in restricted cases. First, we solve an open problem showing that Temporal Arborescence Reconfiguration is NP-hard for two timestamps. Then we show that even if the two temporal arborescences differ only by two arcs, then the problem is not approximable within factor bln|V(D)|, for any constant 0 < b < 1, where V(D) is the set of vertices of the temporal arborescences. Finally, we prove that Temporal Arborescence Reconfiguration is W[1]-hard when parameterized by the number of arc flips needed to transform one temporal arborescence into the other.

Riccardo Dondi and Manuel Lafond. On the Complexity of Temporal Arborescence Reconfiguration. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dondi_et_al:LIPIcs.SAND.2024.10, author = {Dondi, Riccardo and Lafond, Manuel}, title = {{On the Complexity of Temporal Arborescence Reconfiguration}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.10}, URN = {urn:nbn:de:0030-drops-198888}, doi = {10.4230/LIPIcs.SAND.2024.10}, annote = {Keywords: Arborescence, Temporal Graphs, Graph Algorithms, Parameterized Complexity, Approximation Complexity} }

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**Published in:** LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Different variants of Vertex Cover have recently garnered attention in the context of temporal graphs. One of these variants is motivated by the need to summarize timeline activities in social networks. Here, the activities of individual vertices, representing users, are characterized by time intervals. In this paper, we explore a scenario where the temporal span of each vertex’s activity interval is bounded by an integer 𝓁, and the objective is to maximize the number of (temporal) edges that are covered. We establish the APX-hardness of this problem and the NP-hardness of the corresponding decision problem, even under the restricted condition where the temporal domain comprises only two timestamps and each edge appears at most once. Subsequently, we delve into the parameterized complexity of the problem, offering two fixed-parameter algorithms parameterized by: (i) the number k of temporal edges covered by the solution, and (ii) the number h of temporal edges not covered by the solution. Finally, we present a polynomial-time approximation algorithm achieving a factor of 3/4.

Riccardo Dondi, Fabrizio Montecchiani, Giacomo Ortali, Tommaso Piselli, and Alessandra Tappini. Partial Temporal Vertex Cover with Bounded Activity Intervals. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dondi_et_al:LIPIcs.SAND.2024.11, author = {Dondi, Riccardo and Montecchiani, Fabrizio and Ortali, Giacomo and Piselli, Tommaso and Tappini, Alessandra}, title = {{Partial Temporal Vertex Cover with Bounded Activity Intervals}}, booktitle = {3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)}, pages = {11:1--11:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-315-7}, ISSN = {1868-8969}, year = {2024}, volume = {292}, editor = {Casteigts, Arnaud and Kuhn, Fabian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.11}, URN = {urn:nbn:de:0030-drops-198892}, doi = {10.4230/LIPIcs.SAND.2024.11}, annote = {Keywords: Temporal Graphs, Temporal Vertex Cover, Parameterized Complexity, Approximation Algorithms} }

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

Several classical combinatorial problems have been considered and analysed on temporal graphs. Recently, a variant of Vertex Cover on temporal graphs, called MinTimelineCover, has been introduced to summarize timeline activities in social networks. The problem asks to cover every temporal edge while minimizing the total span of the vertices (where the span of a vertex is the length of the timestamp interval it must remain active in). While the problem has been shown to be NP-hard even in very restricted cases, its parameterized complexity has not been fully understood. The problem is known to be in FPT under the span parameter only for graphs with two timestamps, but the parameterized complexity for the general case is open. We settle this open problem by giving an FPT algorithm that is based on a combination of iterative compression and a reduction to the Digraph Pair Cut problem, a powerful problem that has received significant attention recently.

Riccardo Dondi and Manuel Lafond. An FPT Algorithm for Temporal Graph Untangling. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dondi_et_al:LIPIcs.IPEC.2023.12, author = {Dondi, Riccardo and Lafond, Manuel}, title = {{An FPT Algorithm for Temporal Graph Untangling}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {12:1--12:16}, 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.2023.12}, URN = {urn:nbn:de:0030-drops-194311}, doi = {10.4230/LIPIcs.IPEC.2023.12}, annote = {Keywords: Temporal Graphs, Vertex Cover, Graph Algorithms, Parameterized Complexity} }

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

Longest Run Subsequence is a problem introduced recently in the context of the scaffolding phase of genome assembly (Schrinner et al., WABI 2020). The problem asks for a maximum length subsequence of a given string that contains at most one run for each symbol (a run is a maximum substring of consecutive identical symbols). The problem has been shown to be NP-hard and to be fixed-parameter tractable when the parameter is the size of the alphabet on which the input string is defined. In this paper we further investigate the complexity of the problem and we show that it is fixed-parameter tractable when it is parameterized by the number of runs in a solution, a smaller parameter. Moreover, we investigate the kernelization complexity of Longest Run Subsequence and we prove that it does not admit a polynomial kernel when parameterized by the size of the alphabet or by the number of runs. Finally, we consider the restriction of Longest Run Subsequence when each symbol has at most two occurrences in the input string and we show that it is APX-hard.

Riccardo Dondi and Florian Sikora. The Longest Run Subsequence Problem: Further Complexity Results. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{dondi_et_al:LIPIcs.CPM.2021.14, author = {Dondi, Riccardo and Sikora, Florian}, title = {{The Longest Run Subsequence Problem: Further Complexity Results}}, booktitle = {32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021)}, pages = {14:1--14:15}, 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.2021.14}, URN = {urn:nbn:de:0030-drops-139652}, doi = {10.4230/LIPIcs.CPM.2021.14}, annote = {Keywords: Parameterized complexity, Kernelization, Approximation Hardness, Longest Subsequence} }

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**Published in:** LIPIcs, Volume 113, 18th International Workshop on Algorithms in Bioinformatics (WABI 2018)

Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families is interconnected. In this paper, we extend a previous approach to reconcile a set of gene trees with a species tree based on segmental macro-evolutionary events, where segmental duplication events and losses are associated with cost delta and lambda, respectively. We show that the problem is polynomial-time solvable when delta <= lambda (via LCA-mapping), while if delta > lambda the problem is NP-hard, even when lambda = 0 and a single gene tree is given, solving a long standing open problem on the complexity of the reconciliation problem. On the positive side, we give a fixed-parameter algorithm for the problem, where the parameters are delta/lambda and the number d of segmental duplications, of time complexity O(ceil[delta/lambda]^d * n * delta/lambda). Finally, we demonstrate the usefulness of this algorithm on two previously studied real datasets: we first show that our method can be used to confirm or refute hypothetical segmental duplications on a set of 16 eukaryotes, then show how we can detect whole genome duplications in yeast genomes.

Riccardo Dondi, Manuel Lafond, and Celine Scornavacca. Reconciling Multiple Genes Trees via Segmental Duplications and Losses. In 18th International Workshop on Algorithms in Bioinformatics (WABI 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 113, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{dondi_et_al:LIPIcs.WABI.2018.5, author = {Dondi, Riccardo and Lafond, Manuel and Scornavacca, Celine}, title = {{Reconciling Multiple Genes Trees via Segmental Duplications and Losses}}, booktitle = {18th International Workshop on Algorithms in Bioinformatics (WABI 2018)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-082-8}, ISSN = {1868-8969}, year = {2018}, volume = {113}, editor = {Parida, Laxmi and Ukkonen, Esko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2018.5}, URN = {urn:nbn:de:0030-drops-93079}, doi = {10.4230/LIPIcs.WABI.2018.5}, annote = {Keywords: Gene trees/species tree reconciliation, phylogenetics, computational complexity, fixed-parameter algorithms} }

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

Inspired by a recent approach for genome reconstruction from incomplete data, we consider a variant of the longest common subsequence problem for the comparison of two sequences, one of which is incomplete, i.e. it has some missing elements. The new combinatorial problem, called Longest Filled Common Subsequence, given two sequences A and B, and a multiset M of symbols missing in B, asks for a sequence B* obtained by inserting the symbols of M into B so that B* induces a common subsequence with A of maximum length. First, we investigate the computational and approximation complexity of the problem and we show that it is NP-hard and APX-hard when A contains at most two occurrences of each symbol. Then, we give a 3/5 approximation algorithm for the problem. Finally, we present a fixed-parameter algorithm, when the problem is parameterized by the number of symbols inserted in B that "match" symbols of A.

Mauro Castelli, Riccardo Dondi, Giancarlo Mauri, and Italo Zoppis. The Longest Filled Common Subsequence Problem. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{castelli_et_al:LIPIcs.CPM.2017.14, author = {Castelli, Mauro and Dondi, Riccardo and Mauri, Giancarlo and Zoppis, Italo}, title = {{The Longest Filled Common Subsequence Problem}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {14:1--14:13}, 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.2017.14}, URN = {urn:nbn:de:0030-drops-73293}, doi = {10.4230/LIPIcs.CPM.2017.14}, annote = {Keywords: longest common subsequence, approximation algorithms, computational complexity, fixed-parameter algorithms} }

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