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Approximability of Longest Run Subsequence and Complementary Minimization Problems

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Názov: Approximability of Longest Run Subsequence and Complementary Minimization Problems
Autori: Asahiro, Yuichi, Gong, Mingyang, Jansson, Jesper, Lin, Guohui, Lu, Sichen, Miyano, Eiji, Ono, Hirotaka, Saitoh, Toshiki, Tanaka, Shunichi
Prispievatelia: Yuichi Asahiro and Mingyang Gong and Jesper Jansson and Guohui Lin and Sichen Lu and Eiji Miyano and Hirotaka Ono and Toshiki Saitoh and Shunichi Tanaka
Informácie o vydavateľovi: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2025.
Rok vydania: 2025
Predmety: minimum run subsequence deletion, Longest run subsequence, ddc:004, approximation algorithm
Popis: We study the polynomial-time approximability of the Longest Run Subsequence problem (LRS for short) and its complementary minimization variant Minimum Run Subsequence Deletion problem (MRSD for short). For a string S = s₁ ⋯ s_n over an alphabet Σ, a subsequence S' of S is S' = s_{i₁} ⋯ s_{i_p}, such that 1 ≤ i₁ < i₂ < … < i_p ≤ |S|. 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 subsequence of S in which every symbol σ ∈ Σ occurs in at most one run. The co-subsequence ̅{S'} of the subsequence S' = s_{i₁} ⋯ s_{i_p} in S is the subsequence obtained by deleting all the characters in S' from S, i.e., ̅{S'} = s_{j₁} ⋯ s_{j_{n-p}} such that j₁ < j₂ < … < j_{n-p} and {j₁, …, j_{n-p}} = {1, …, n}⧵ {i₁, …, i_p}. Given a string S, the goal of LRS (resp., MRSD) is to find a run subsequence S^* of S such that the length |S^*| is maximized (resp., the number | ̅{S^*}| of deleted symbols from S is minimized) over all the run subsequences of S. Let k be the maximum number of symbol occurrences in the input S. It is known that LRS and MRSD are APX-hard even if k = 2. In this paper, we show that LRS can be approximated in polynomial time within factors of (k+2)/3 for k = 2 or 3, and 2(k+1)/5 for every k ≥ 4. Furthermore, we show that MRSD can be approximated in linear time within a factor of (k+4)/4 if k is even and (k+3)/4 if k is odd.
Druh dokumentu: Conference object
Popis súboru: application/pdf
Jazyk: English
DOI: 10.4230/lipics.wabi.2025.3
Prístupová URL adresa: https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2025.3
Rights: CC BY
Prístupové číslo: edsair.od......1814..d8060764afad6bb7fbc75fcecfc6eabf
Databáza: OpenAIRE
Popis
Abstrakt:We study the polynomial-time approximability of the Longest Run Subsequence problem (LRS for short) and its complementary minimization variant Minimum Run Subsequence Deletion problem (MRSD for short). For a string S = s₁ ⋯ s_n over an alphabet Σ, a subsequence S' of S is S' = s_{i₁} ⋯ s_{i_p}, such that 1 ≤ i₁ < i₂ < … < i_p ≤ |S|. 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 subsequence of S in which every symbol σ ∈ Σ occurs in at most one run. The co-subsequence ̅{S'} of the subsequence S' = s_{i₁} ⋯ s_{i_p} in S is the subsequence obtained by deleting all the characters in S' from S, i.e., ̅{S'} = s_{j₁} ⋯ s_{j_{n-p}} such that j₁ < j₂ < … < j_{n-p} and {j₁, …, j_{n-p}} = {1, …, n}⧵ {i₁, …, i_p}. Given a string S, the goal of LRS (resp., MRSD) is to find a run subsequence S^* of S such that the length |S^*| is maximized (resp., the number | ̅{S^*}| of deleted symbols from S is minimized) over all the run subsequences of S. Let k be the maximum number of symbol occurrences in the input S. It is known that LRS and MRSD are APX-hard even if k = 2. In this paper, we show that LRS can be approximated in polynomial time within factors of (k+2)/3 for k = 2 or 3, and 2(k+1)/5 for every k ≥ 4. Furthermore, we show that MRSD can be approximated in linear time within a factor of (k+4)/4 if k is even and (k+3)/4 if k is odd.
DOI:10.4230/lipics.wabi.2025.3