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Approximability and Fixed-Parameter Tractability for the Exemplar Genomic Distance Problems

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Book cover Theory and Applications of Models of Computation (TAMC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5532))

Abstract

In this paper, we present a survey of the approximability and fixed-parameter tractability results for some Exemplar Genomic Distance problems. We mainly focus on three problems: the exemplar breakpoint distance problem and its complement (i.e., the exemplar non-breaking similarity or the exemplar adjacency number problem), and the maximal strip recovery (MSR) problem. The following results hold for the simplest case between only two genomes (genomic maps) \({\cal G}\) and \({\cal H}\), each containing only one sequence of genes (gene markers), possibly with repetitions.

  1. 1

    For the general Exemplar Breakpoint Distance problem, it was shown that deciding if the optimal solution value of some given instance is zero is NP-hard. This implies that the problem does not admit any approximation, neither any FPT algorithm, unless P=NP. In fact, this result holds even when a gene appears in \({\cal G}\) (\({\cal H}\)) at most two times.

  2. 1

    For the Exemplar Non-breaking Similarity problem, it was shown that the problem is linearly reducible from Independent Set. Hence, it does not admit any factor-O(n ε) approximation unless P=NP and it is W[1]-complete (loosely speaking, there is no way to obtain an O(n o(k)) time exact algorithm unless FPT=W[1], here k is the optimal solution value of the problem).

  3. 1

    For the MSR problem, after quite a lot of struggle, we recently showed that the problem is NP-complete. On the other hand, the problem was previously known to have a factor-4 approximation and we showed recently that it admits a simple FPT algorithm which runs in O(22.73k n + n 2) time, where k is the optimal solution value of the problem.

This research is partially supported by NSF, NSERC, Louisiana Board of Regents under contract number LEQSF(2004-07)-RD-A-35, and MSU-Bozeman’s Short-Term Professional Development Leave Program.

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Authors and Affiliations

  1. Department of Computer Science, Montana State University, Bozeman, MT 59717-3880, USA

    Binhai Zhu

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  1. Binhai Zhu
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Editors and Affiliations

  1. Department of Computer Science and Engineering, Texas A&M University, College Station, 77843, Texas, USA

    Jianer Chen

  2. School of Mathematics, University of Leeds, LS2 9JT, U.K.

    S. Barry Cooper

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Zhu, B. (2009). Approximability and Fixed-Parameter Tractability for the Exemplar Genomic Distance Problems. In: Chen, J., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2009. Lecture Notes in Computer Science, vol 5532. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02017-9_10

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  • DOI: https://doi.org/10.1007/978-3-642-02017-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02016-2

  • Online ISBN: 978-3-642-02017-9

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