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Consensus Strings with Small Maximum Distance and Small Distance Sum

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Abstract

The parameterised complexity of various consensus string problems (Closest String, Closest Substring, Closest String with Outliers) is investigated in a more general setting, i. e., with a bound on the maximum Hamming distance and a bound on the sum of Hamming distances between solution and input strings. We completely settle the parameterised complexity of these generalised variants of Closest String and Closest Substring, and partly for Closest String with Outliers; in addition, we answer some open questions from the literature regarding the classical problem variants with only one distance bound. Finally, we investigate the question of polynomial kernels and respective lower bounds.

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Notes

  1. Note that we slightly abuse notation with respect to the subset relation: for a multi-set A and a set B, \(A \subseteq B\) means that \(A' \subseteq B\), where \(A'\) is the set obtained from A by deleting duplicates; for multi-sets A, B, \(A \subseteq B\) is defined as usual. Moreover, whenever it is clear from the context that we talk about multi-sets, we also simply use the term set.

  2. For a corresponding table of the already known results for \({{\,\mathrm{(\textsf {r})-\textsc {CloseSubstr}}\,}}\), see, e. g., [29, Table 1].

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Acknowledgements

We wish to thank the anonymous referees for valuable feedback that improved the readability of this paper.

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Authors and Affiliations

  1. Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEM, 77454, Marne-la-Vallée, France

    Laurent Bulteau

  2. Fachbereich 4 – Abteilung Informatikwissenschaften, Universität Trier, 54286, Trier, Germany

    Markus L. Schmid

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  1. Laurent Bulteau
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Correspondence to Markus L. Schmid.

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Bulteau, L., Schmid, M.L. Consensus Strings with Small Maximum Distance and Small Distance Sum. Algorithmica 82, 1378–1409 (2020). https://doi.org/10.1007/s00453-019-00647-9

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