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

We study the approximability of the Longest Run Subsequence problem (LRS for short). For a string S = s_1 ⋯ s_n over an alphabet Σ, a run of a symbol σ ∈ Σ in S is a maximal substring of consecutive occurrences of σ. A run subsequence S' of S is a sequence in which every symbol σ ∈ Σ occurs in at most one run. Given a string S, the goal of LRS is to find a longest run subsequence S^* of S such that the length |S^*| is maximized over all the run subsequences of S. It is known that LRS is APX-hard even if each symbol has at most two occurrences in the input string, and that LRS admits a polynomial-time k-approximation algorithm if the number of occurrences of every symbol in the input string is bounded by k. In this paper, we design a polynomial-time (k+1)/2-approximation algorithm for LRS under the k-occurrence constraint on input strings. For the case k = 2, we further improve the approximation ratio from 3/2 to 4/3.

Yuichi Asahiro, Hiroshi Eto, Mingyang Gong, Jesper Jansson, Guohui Lin, Eiji Miyano, Hirotaka Ono, and Shunichi Tanaka. Approximation Algorithms for the Longest Run Subsequence Problem. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 2:1-2:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{asahiro_et_al:LIPIcs.CPM.2023.2, author = {Asahiro, Yuichi and Eto, Hiroshi and Gong, Mingyang and Jansson, Jesper and Lin, Guohui and Miyano, Eiji and Ono, Hirotaka and Tanaka, Shunichi}, title = {{Approximation Algorithms for the Longest Run Subsequence Problem}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {2:1--2:12}, 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.2023.2}, URN = {urn:nbn:de:0030-drops-179560}, doi = {10.4230/LIPIcs.CPM.2023.2}, annote = {Keywords: Longest run subsequence problem, bounded occurrence, approximation algorithm} }

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

A multi-labelled tree (or MUL-tree) is a rooted tree leaf-labelled by a set of labels, where each label may appear more than once in the tree. We consider the MUL-tree Set Pruning for Consistency problem (MULSETPC), which takes as input a set of MUL-trees and asks whether there exists a perfect pruning of each MUL-tree that results in a consistent set of single-labelled trees. MULSETPC was proven to be NP-complete by Gascon et al. when the MUL-trees are binary, each leaf label is used at most three times, and the number of MUL-trees is unbounded. To determine the computational complexity of the problem when the number of MUL-trees is constant was left as an open problem.
Here, we resolve this question by proving a much stronger result, namely that MULSETPC is NP-complete even when there are only two MUL-trees, every leaf label is used at most twice, and every MUL-tree is either binary or has constant height. Furthermore, we introduce an extension of MULSETPC that we call MULSETPComp, which replaces the notion of consistency with compatibility, and prove that MULSETPComp is NP-complete even when there are only two MUL-trees, every leaf label is used at most thrice, and every MUL-tree has constant height. Finally, we present a polynomial-time algorithm for instances of MULSETPC with a constant number of binary MUL-trees, in the special case where every leaf label occurs exactly once in at least one MUL-tree.

Christopher Hampson, Daniel J. Harvey, Costas S. Iliopoulos, Jesper Jansson, Zara Lim, and Wing-Kin Sung. MUL-Tree Pruning for Consistency and Compatibility. In 34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 259, pp. 14:1-14:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hampson_et_al:LIPIcs.CPM.2023.14, author = {Hampson, Christopher and Harvey, Daniel J. and Iliopoulos, Costas S. and Jansson, Jesper and Lim, Zara and Sung, Wing-Kin}, title = {{MUL-Tree Pruning for Consistency and Compatibility}}, booktitle = {34th Annual Symposium on Combinatorial Pattern Matching (CPM 2023)}, pages = {14:1--14:18}, 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.2023.14}, URN = {urn:nbn:de:0030-drops-179682}, doi = {10.4230/LIPIcs.CPM.2023.14}, annote = {Keywords: multi-labelled tree, phylogenetic tree, consistent, compatible, pruning, algorithm, NP-complete} }

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

The problem of computing the longest common subsequence of two sequences (LCS for short) is a classical and fundamental problem in computer science. In this paper, we study four variants of LCS: the Repetition-Bounded Longest Common Subsequence problem (RBLCS) [Yuichi Asahiro et al., 2020], the Multiset-Restricted Common Subsequence problem (MRCS) [Radu Stefan Mincu and Alexandru Popa, 2018], the Two-Side-Filled Longest Common Subsequence problem (2FLCS), and the One-Side-Filled Longest Common Subsequence problem (1FLCS) [Mauro Castelli et al., 2017; Mauro Castelli et al., 2019]. Although the original LCS can be solved in polynomial time, all these four variants are known to be NP-hard. Recently, an exact, O(1.44225ⁿ)-time, dynamic programming (DP)-based algorithm for RBLCS was proposed [Yuichi Asahiro et al., 2020], where the two input sequences have lengths n and poly(n). We first establish that each of MRCS, 1FLCS, and 2FLCS is polynomially equivalent to RBLCS. Then, we design a refined DP-based algorithm for RBLCS that runs in O(1.41422ⁿ) time, which implies that MRCS, 1FLCS, and 2FLCS can also be solved in O(1.41422ⁿ) time. Finally, we give a polynomial-time 2-approximation algorithm for 2FLCS.

Yuichi Asahiro, Jesper Jansson, Guohui Lin, Eiji Miyano, Hirotaka Ono, and Tadatoshi Utashima. Polynomial-Time Equivalences and Refined Algorithms for Longest Common Subsequence Variants. In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, pp. 15:1-15:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{asahiro_et_al:LIPIcs.CPM.2022.15, author = {Asahiro, Yuichi and Jansson, Jesper and Lin, Guohui and Miyano, Eiji and Ono, Hirotaka and Utashima, Tadatoshi}, title = {{Polynomial-Time Equivalences and Refined Algorithms for Longest Common Subsequence Variants}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {15:1--15:17}, 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.2022.15}, URN = {urn:nbn:de:0030-drops-161424}, doi = {10.4230/LIPIcs.CPM.2022.15}, annote = {Keywords: Repetition-bounded longest common subsequence problem, multiset restricted longest common subsequence problem, one-side-filled longest common subsequence problem, two-side-filled longest common subsequence problem, exact algorithms, and approximation algorithms} }

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**Published in:** LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size n of the population is embedded in the transition function. Their efficiency is expressed in terms of the number of states utilized by agents, the size of the range from which the labels are drawn, and the expected number of interactions required by our solutions. Our primary goal is to provide efficient protocols for this fundamental problem complemented with tight lower bounds in all the three aspects. W.h.p. (with high probability), our labeling protocols are silent, i.e., eventually each agent reaches its final state and remains in it forever, and they are safe, i.e., never update the label assigned to any single agent. We first present a silent w.h.p. and safe labeling protocol that draws labels from the range [1,2n]. Both the number of interactions required and the number of states used by the protocol are asymptotically optimal, i.e., O(n log n) w.h.p. and O(n), respectively. Next, we present a generalization of the protocol, where the range of assigned labels is [1,(1+ε) n]. The generalized protocol requires O(n log n / ε) interactions in order to complete the assignment of distinct labels from [1,(1+ε) n] to the n agents, w.h.p. It is also silent w.h.p. and safe, and uses (2+ε)n+O(n^c) states, for any positive c < 1. On the other hand, we consider the so-called pool labeling protocols that include our fast protocols. We show that the expected number of interactions required by any pool protocol is ≥ (n²)/(r+1), when the labels range is 1,… , n+r < 2n. Furthermore, we provide a protocol which uses only n+5√ n +O(n^c) states, for any c < 1, and draws labels from the range 1,… ,n. The expected number of interactions required by the protocol is O(n³). Once a unique leader is elected it produces a valid labeling and it is silent and safe. On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability > 1-(1/n), uses ≥ n+√{(n-1)/2}-1 states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Finally, we show that for any silent and safe labeling protocol utilizing n+t < 2n states, the expected number of interactions required to achieve a valid labeling is ≥ (n²)/(t+1).

Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas. Efficient Assignment of Identities in Anonymous Populations. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 12:1-12:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.OPODIS.2021.12, author = {G\k{a}sieniec, Leszek and Jansson, Jesper and Levcopoulos, Christos and Lingas, Andrzej}, title = {{Efficient Assignment of Identities in Anonymous Populations}}, booktitle = {25th International Conference on Principles of Distributed Systems (OPODIS 2021)}, pages = {12:1--12:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-219-8}, ISSN = {1868-8969}, year = {2022}, volume = {217}, editor = {Bramas, Quentin and Gramoli, Vincent and Milani, Alessia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.12}, URN = {urn:nbn:de:0030-drops-157871}, doi = {10.4230/LIPIcs.OPODIS.2021.12}, annote = {Keywords: population protocol, state efficiency, time efficiency, one-way epidemics, leader election, agent identities} }

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**Published in:** LIPIcs, Volume 143, 19th International Workshop on Algorithms in Bioinformatics (WABI 2019)

We combine two fundamental, previously studied optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency (MAXRTC) and minimally resolved supertree (MINRS) into a new problem, which we call q-maximum rooted triplets consistency (q-MAXRTC). The input to our new problem is a set R of resolved triplets (rooted, binary phylogenetic trees with three leaves each) and the objective is to find a phylogenetic tree with exactly q internal nodes that contains the largest possible number of triplets from R. We first prove that q-MAXRTC is NP-hard even to approximate within a constant ratio for every fixed q >= 2, and then develop various polynomial-time approximation algorithms for different values of q. Next, we show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much triplet branching information. As an extreme example, we show that allowing only nine internal nodes is still sufficient to capture on average 80% of the rooted triplets from some recently published trees, each having between 760 and 3081 internal nodes. Finally, to demonstrate the algorithmic advantage of using trees with few internal nodes, we propose a new algorithm for computing the rooted triplet distance between two phylogenetic trees over a leaf label set of size n that runs in O(q n) time, where q is the number of internal nodes in the smaller tree, and is therefore faster than the currently best algorithms for the problem (with O(n log n) time complexity [SODA 2013, ESA 2017]) whenever q = o(log n).

Jesper Jansson, Konstantinos Mampentzidis, and Sandhya T. P.. Building a Small and Informative Phylogenetic Supertree. In 19th International Workshop on Algorithms in Bioinformatics (WABI 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 143, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jansson_et_al:LIPIcs.WABI.2019.1, author = {Jansson, Jesper and Mampentzidis, Konstantinos and T. P., Sandhya}, title = {{Building a Small and Informative Phylogenetic Supertree}}, booktitle = {19th International Workshop on Algorithms in Bioinformatics (WABI 2019)}, pages = {1:1--1:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-123-8}, ISSN = {1868-8969}, year = {2019}, volume = {143}, editor = {Huber, Katharina T. and Gusfield, Dan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2019.1}, URN = {urn:nbn:de:0030-drops-110316}, doi = {10.4230/LIPIcs.WABI.2019.1}, annote = {Keywords: phylogenetic tree, supertree, rooted triplet, approximation algorithm} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

The tree inclusion problem is, given two node-labeled trees P and T (the "pattern tree" and the "text tree"), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is from 1995 and runs in O(poly(m,n) * 2^{2d}) = O^*(2^{2d}) time, where m and n are the sizes of the pattern and text trees, respectively, and d is the maximum outdegree of the pattern tree. Here, we develop a new algorithm that improves the exponent 2d to d by considering a particular type of ancestor-descendant relationships and applying dynamic programming, thus reducing the time complexity to O^*(2^d). We then study restricted variants of the unordered tree inclusion problem where the number of occurrences of different node labels and/or the input trees' heights are bounded. We show that although the problem remains NP-hard in many such cases, it can be solved in polynomial time for c = 2 and in O^*(1.8^d) time for c = 3 if the leaves of P are distinctly labeled and each label occurs at most c times in T. We also present a randomized O^*(1.883^d)-time algorithm for the case that the heights of P and T are one and two, respectively.

Tatsuya Akutsu, Jesper Jansson, Ruiming Li, Atsuhiro Takasu, and Takeyuki Tamura. New and Improved Algorithms for Unordered Tree Inclusion. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 27:1-27:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{akutsu_et_al:LIPIcs.ISAAC.2018.27, author = {Akutsu, Tatsuya and Jansson, Jesper and Li, Ruiming and Takasu, Atsuhiro and Tamura, Takeyuki}, title = {{New and Improved Algorithms for Unordered Tree Inclusion}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {27:1--27:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.27}, URN = {urn:nbn:de:0030-drops-99752}, doi = {10.4230/LIPIcs.ISAAC.2018.27}, annote = {Keywords: parameterized algorithms, tree inclusion, unordered trees, dynamic programming} }

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

The problem of constructing a minimally resolved phylogenetic supertree (i.e., having the smallest possible number of internal nodes) that contains all of the rooted triplets from a consistent set R is known to be NP-hard. In this paper, we prove that constructing a phylogenetic tree consistent with R that contains the minimum number of additional rooted triplets is also NP-hard, and develop exact, exponential-time algorithms for both problems. The new algorithms are applied to construct two variants of the local consensus tree;
for any set S of phylogenetic trees over some leaf label set L,
this gives a minimal phylogenetic tree over L that contains every
rooted triplet present in all trees in S, where ``minimal'' means either having the smallest possible number of internal nodes or
the smallest possible number of rooted triplets. The second variant generalizes the RV-II tree, introduced by Kannan, Warnow, and Yooseph in 1998.

Jesper Jansson and Wing-Kin Sung. Minimal Phylogenetic Supertrees and Local Consensus Trees. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 53:1-53:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{jansson_et_al:LIPIcs.MFCS.2016.53, author = {Jansson, Jesper and Sung, Wing-Kin}, title = {{Minimal Phylogenetic Supertrees and Local Consensus Trees}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {53:1--53:14}, 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.2016.53}, URN = {urn:nbn:de:0030-drops-64653}, doi = {10.4230/LIPIcs.MFCS.2016.53}, annote = {Keywords: phylogenetic tree, rooted triplet, local consensus, minimal supertree, computational complexity, bioinformatics} }

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

This paper presents a fast algorithm for finding the Adams consensus tree of a set of conflicting phylogenetic trees with identical leaf labels, for the first time improving the time complexity of a widely used algorithm invented by Adams in 1972 [1]. Our algorithm applies
the centroid path decomposition technique [9] in a new way to traverse the input trees' centroid paths in unison, and runs in O(k n \log n) time, where k is the number of input trees and n is the size of the leaf label set. (In comparison, the old algorithm from 1972 has a worst-case running time of O(k n^2).) For the special case of k = 2, an even faster algorithm running in O(n \cdot \frac{\log n}{\log\log n}) time is provided, which relies on an extension of the wavelet tree-based technique by Bose et al. [6] for orthogonal range counting on a grid.
Our extended wavelet tree data structure also supports truncated
range maximum queries efficiently and may be of independent interest to algorithm designers.

Jesper Jansson, Zhaoxian Li, and Wing-Kin Sung. On Finding the Adams Consensus Tree. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 487-499, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{jansson_et_al:LIPIcs.STACS.2015.487, author = {Jansson, Jesper and Li, Zhaoxian and Sung, Wing-Kin}, title = {{On Finding the Adams Consensus Tree}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {487--499}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.487}, URN = {urn:nbn:de:0030-drops-49364}, doi = {10.4230/LIPIcs.STACS.2015.487}, annote = {Keywords: phylogenetic tree, Adams consensus, centroid path, wavelet tree} }

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