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Consensus Patterns (Probably) Has no EPTAS

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Book cover Algorithms - ESA 2015

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9294))

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Abstract

Given n length-L strings S = {s 1, …, s n } over a constant size alphabet Σ together with an integer ℓ, where ℓ ≤ L, the objective of Consensus Patterns is to find a length-ℓ string s, a substring t i of each s i in S such that ∑  ∀ i d(t i , s) is minimized. Here d(x, y) denotes the Hamming distance between the two strings x and y. Consensus Patterns admits a PTAS  [Li et al., JCSS 2002] is fixed parameter tractable when parameterized by the objective function value [Marx, SICOMP 2008], and although it is a well-studied problem, improvement of the PTAS to an EPTAS seemed elusive. We prove that Consensus Patterns does not admit an EPTAS unless FPT=W[1], answering an open problem from [Fellows et al., STACS 2002, Combinatorica 2006]. To the best of our knowledge, Consensus Patterns is the first problem that admits a PTAS, and is fixed parameter tractable when parameterized by the value of the objective function but does not admit an EPTAS under plausible complexity assumptions. The proof of our hardness of approximation result combines parameterized reductions and gap preserving reductions in a novel manner.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Colorado State University in Fort Collins, Fort Collins, USA

    Christina Boucher

  2. Department of Computer Science and Engineering, University of California at San Diego, San Diego, USA

    Christine Lo

  3. Department of Informatics, University of Bergen, Bergen, Norway

    Daniel Lokshantov

Authors
  1. Christina Boucher
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  2. Christine Lo
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  3. Daniel Lokshantov
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Corresponding author

Correspondence to Christina Boucher .

Editor information

Editors and Affiliations

  1. University of Technology, Eindhoven, The Netherlands

    Nikhil Bansal

  2. Sapienza University of Rome, Rome, Italy

    Irene Finocchi

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Boucher, C., Lo, C., Lokshantov, D. (2015). Consensus Patterns (Probably) Has no EPTAS. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_21

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  • DOI: https://doi.org/10.1007/978-3-662-48350-3_21

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48349-7

  • Online ISBN: 978-3-662-48350-3

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