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Approximability of Longest Run Subsequence and Complementary Minimization Problems

Authors Yuichi Asahiro , Mingyang Gong , Jesper Jansson , Guohui Lin , Sichen Lu, Eiji Miyano , Hirotaka Ono , Toshiki Saitoh , Shunichi Tanaka

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Longest run subsequence
  • minimum run subsequence deletion
  • approximation algorithm

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Abstract

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.

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Yuichi Asahiro, Mingyang Gong, Jesper Jansson, Guohui Lin, Sichen Lu, Eiji Miyano, Hirotaka Ono, Toshiki Saitoh, and Shunichi Tanaka. Approximability of Longest Run Subsequence and Complementary Minimization Problems. In 25th International Conference on Algorithms for Bioinformatics (WABI 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 344, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WABI.2025.3

Author Details

Yuichi Asahiro
  • Kyushu Sangyo University, Fukuoka, Japan
Mingyang Gong
  • University of Alberta, Edmonton, Canada
Jesper Jansson
  • Kyoto University, Kyoto, Japan
Guohui Lin
  • University of Alberta, Edmonton, Canada
Sichen Lu
  • University of Alberta, Edmonton, Canada
Eiji Miyano
  • Kyushu Institute of Technology, Iizuka, Japan
Hirotaka Ono
  • Nagoya University, Nagoya, Japan
Toshiki Saitoh
  • Kyushu Institute of Technology, Iizuka, Japan
Shunichi Tanaka
  • Kyushu Institute of Technology, Iizuka, Japan

Funding

The work was partially supported by the NSERC Canada, JSPS KAKENHI Grant Numbers JP20H05967, JP22H00513, JP22K11915, JP24H00697, JP24K02898, JP24K02902, JP24K14827, and JP25K03077, and CRONOS Grant Number JPMJCS24K2.

References

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