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Approximation and Fixed Parameter Algorithms for the Approximate Cover Problem

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String Processing and Information Retrieval (SPIRE 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14240))

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Abstract

Amir et al. (CPM 2017) introduce the approximate string cover problem (ACP) motivated by applications including molecular biology, coding, automata theory, formal language theory and combinatorics. A cover of a string T is a string C for which every letter of T lies within some occurrence of C. The input of the ACP consists of a string T and the goal is to find a string C of length less than the length of T that covers a string \(T'\), which is as close to T as possible (under some predefined distance). Amir et al. study this problem for the Hamming distance and show that it is NP-hard.

In this paper we continue the work of Amir et al. and show the following results for the cover length relaxation of the ACP. After observing that the NP-hardness proof by Amir et al. (CPM 2017, TCS 2019) suffers from several lapses, we propose an amendment to the proof. We then introduce an approximation algorithm for a variant of the ACP, in which we aim to maximize the length of the input string minus the distance to the string covered by the approximate cover returned by the algorithm. This problem is naturally as hard as the ACP. We prove an asymptotic approximation ratio of \(\mathcal {O}(\sqrt{|T|})\), where |T| is the size of the input string. Finally, we present an FPT algorithm with respect to the alphabet size and the size of the cover based on a dynamic programming framework.

This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2021-0253, within PNCDI III. We also thank the PHC Brincusi project between Univ. of Bordeaux and Univ. of Bucharest that facilitated the bilateral visits leading to this work.

M. Raffinot—Currently at Apple.

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Notes

  1. 1.

    In the original construction \(y_4=x_0\), \(y_3=x_{-1}\), \(y_2=x_{N+1}\) and \(y_1=x_{N+2}\).

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Authors and Affiliations

  1. CNRS, Bordeaux INP, LaBRI, UMR 5800, Univ. Bordeaux, 33400, Talence, France

    Guillaume Blin, Mathieu Raffinot & Raluca Uricaru

  2. Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania

    Alexandru Popa

Authors
  1. Guillaume Blin
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  2. Alexandru Popa
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  3. Mathieu Raffinot
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  4. Raluca Uricaru
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Correspondence to Raluca Uricaru .

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Editors and Affiliations

  1. ISTI-CNR, Pisa, Italy

    Franco Maria Nardini

  2. University of Pisa, Pisa, Italy

    Nadia Pisanti

  3. University of Pisa, Pisa, Italy

    Rossano Venturini

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Blin, G., Popa, A., Raffinot, M., Uricaru, R. (2023). Approximation and Fixed Parameter Algorithms for the Approximate Cover Problem. In: Nardini, F.M., Pisanti, N., Venturini, R. (eds) String Processing and Information Retrieval. SPIRE 2023. Lecture Notes in Computer Science, vol 14240. Springer, Cham. https://doi.org/10.1007/978-3-031-43980-3_7

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  • DOI: https://doi.org/10.1007/978-3-031-43980-3_7

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