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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 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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