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Finding Consensus Strings with Small Length Difference between Input and Solution Strings

Finding Consensus Strings with Small Length Difference between Input and Solution Strings The Closest Substring Problem is to decide, for given strings s1, , sk of length at most and numbers m and d, whether there is a length-m string s and length-m substrings si of si, such that s has a Hamming distance of at most d from each si. If we instead require the sum of all the Hamming distances between s and each si to be bounded by d, then it is called the Consensus Patterns Problem. We contribute to the parameterised complexity analysis of these classical NP-hard string problems by investigating the parameter ( m), i.e., the length difference between input and solution strings. For most combinations of ( m) and one of the classical parameters (m, , k, or d), we obtain fixed-parameter tractability. However, even for constant ( m) and constant alphabet size, both problems remain NP-hard. While this follows from known results with respect to the Closest Substring, we need a new reduction in the case of the Consensus Patterns. As a by-product of this reduction, we obtain an exact exponential-time algorithm for both problems, which is based on an alphabet reduction. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computation Theory (TOCT) Association for Computing Machinery

Finding Consensus Strings with Small Length Difference between Input and Solution Strings

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Publisher
Association for Computing Machinery
Copyright
Copyright © 2017 ACM
ISSN
1942-3454
eISSN
1942-3462
DOI
10.1145/3110290
Publisher site
See Article on Publisher Site

Abstract

The Closest Substring Problem is to decide, for given strings s1, , sk of length at most and numbers m and d, whether there is a length-m string s and length-m substrings si of si, such that s has a Hamming distance of at most d from each si. If we instead require the sum of all the Hamming distances between s and each si to be bounded by d, then it is called the Consensus Patterns Problem. We contribute to the parameterised complexity analysis of these classical NP-hard string problems by investigating the parameter ( m), i.e., the length difference between input and solution strings. For most combinations of ( m) and one of the classical parameters (m, , k, or d), we obtain fixed-parameter tractability. However, even for constant ( m) and constant alphabet size, both problems remain NP-hard. While this follows from known results with respect to the Closest Substring, we need a new reduction in the case of the Consensus Patterns. As a by-product of this reduction, we obtain an exact exponential-time algorithm for both problems, which is based on an alphabet reduction.

Journal

ACM Transactions on Computation Theory (TOCT) – Association for Computing Machinery

Published: Sep 18, 2017

Keywords: Parameterised complexity

References

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    Bulteau, L.; Hüffner, F.; Komusiewicz, C.; Niedermeier, R.
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    Fellows, M. R.; Gramm, J.; Niedermeier, R.
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    Flum, J.; Grohe, M.
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    Frances, M.; Litman, A.
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    Garey, M. R.; Johnson, D. S.
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    Gramm, J.; Niedermeier, R.; Rossmanith, P.
  • Closest substring problems with small distances
    Marx, D.
  • Finding consensus strings with small length difference between input and solution strings
    Schmid, M. L.