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Superstrings with multiplicities

Authors Bastien Cazaux, Eric Rivals

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Approximation algorithms analysis
Keywords
  • greedy algorithm
  • approximation
  • overlap
  • cyclic cover
  • APX
  • subset system

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Abstract

A superstring of a set of words P = {s_1, ..., s_p } is a string that contains each word of P as substring. Given P, the well known Shortest Linear Superstring problem (SLS), asks for a shortest superstring of P. In a variant of SLS, called Multi-SLS, each word s_i comes with an integer m(i), its multiplicity, that sets a constraint on its number of occurrences, and the goal is to find a shortest superstring that contains at least m(i) occurrences of s_i. Multi-SLS generalizes SLS and is obviously as hard to solve, but it has been studied only in special cases (with words of length 2 or with a fixed number of words). The approximability of Multi-SLS in the general case remains open. Here, we study the approximability of Multi-SLS and that of the companion problem Multi-SCCS, which asks for a shortest cyclic cover instead of shortest superstring. First, we investigate the approximation of a greedy algorithm for maximizing the compression offered by a superstring or by a cyclic cover: the approximation ratio is 1/2 for Multi-SLS and 1 for Multi-SCCS. Then, we exhibit a linear time approximation algorithm, Concat-Greedy, and show it achieves a ratio of 4 regarding the superstring length. This demonstrates that for both measures Multi-SLS belongs to the class of APX problems.

Cite As Get BibTex

Bastien Cazaux and Eric Rivals. Superstrings with multiplicities. In 29th Annual Symposium on Combinatorial Pattern Matching (CPM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 105, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CPM.2018.21

Author Details

Bastien Cazaux
  • Department of Computer Science, University of Helsinki, Helsinki, Finland
  • , L.I.R.M.M., Université Montpellier, Montpellier, France & Institute of Computational Biology, Montpellier, France
Eric Rivals
  • L.I.R.M.M., Université Montpellier, Montpellier, France & Institute of Computational Biology, Montpellier, France

Funding

This work is supported by the {Institut de Biologie Computationnelle} http://www.ibc-montpellier.fr (ANR-11-BINF-0002) and Défi {MASTODONS C3G} http://www.lirmm.fr/rivals/C3G/ from CNRS.

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