The Longest Run Subsequence Problem: Further Complexity Results

Authors Riccardo Dondi , Florian Sikora

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Author Details

Riccardo Dondi
  • Università degli Studi di Bergamo, Bergamo, Italy
Florian Sikora
  • Université Paris-Dauphine, PSL University, CNRS, LAMSADE, 75016 Paris, France

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Riccardo Dondi and Florian Sikora. The Longest Run Subsequence Problem: Further Complexity Results. In 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 191, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Longest Run Subsequence is a problem introduced recently in the context of the scaffolding phase of genome assembly (Schrinner et al., WABI 2020). The problem asks for a maximum length subsequence of a given string that contains at most one run for each symbol (a run is a maximum substring of consecutive identical symbols). The problem has been shown to be NP-hard and to be fixed-parameter tractable when the parameter is the size of the alphabet on which the input string is defined. In this paper we further investigate the complexity of the problem and we show that it is fixed-parameter tractable when it is parameterized by the number of runs in a solution, a smaller parameter. Moreover, we investigate the kernelization complexity of Longest Run Subsequence and we prove that it does not admit a polynomial kernel when parameterized by the size of the alphabet or by the number of runs. Finally, we consider the restriction of Longest Run Subsequence when each symbol has at most two occurrences in the input string and we show that it is APX-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Graph algorithms analysis
  • Parameterized complexity
  • Kernelization
  • Approximation Hardness
  • Longest Subsequence


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