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# Polynomial-Time Equivalences and Refined Algorithms for Longest Common Subsequence Variants

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## Cite As

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)
https://doi.org/10.4230/LIPIcs.CPM.2022.15

## Abstract

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.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### 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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## References

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