The Uniform Minimum-Ones 2SAT Problem and its Application to Haplotype Classification

Hans-Joachim Böckenhauer; Michal Forišek; Ján Oravec; Björn Steffen; Kathleen Steinhöfel; Monika Steinová

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 44, Issue: 3, page 363-377
  • ISSN: 0988-3754

Abstract

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Analyzing genomic data for finding those gene variations which are responsible for hereditary diseases is one of the great challenges in modern bioinformatics. In many living beings (including the human), every gene is present in two copies, inherited from the two parents, the so-called haplotypes. In this paper, we propose a simple combinatorial model for classifying the set of haplotypes in a population according to their responsibility for a certain genetic disease. This model is based on the minimum-ones 2SAT problem with uniform clauses. The minimum-ones 2SAT problem asks for a satisfying assignment to a satisfiable formula in 2CNF which sets a minimum number of variables to true. This problem is well-known to be 𝒩𝒫 -hard, even in the case where all clauses are uniform, i.e., do not contain a positive and a negative literal. We analyze the approximability and present the first non-trivial exact algorithm for the uniform minimum-ones 2SAT problem with a running time of 𝒪 (1.21061n) on a 2SAT formula with n variables. We also show that the problem is fixed-parameter tractable by showing that our algorithm can be adapted to verify in 𝒪 * (2k) time whether an assignment with at most k true variables exists.

How to cite

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Böckenhauer, Hans-Joachim, et al. "The Uniform Minimum-Ones 2SAT Problem and its Application to Haplotype Classification." RAIRO - Theoretical Informatics and Applications 44.3 (2010): 363-377. <http://eudml.org/doc/250809>.

@article{Böckenhauer2010,
abstract = { Analyzing genomic data for finding those gene variations which are responsible for hereditary diseases is one of the great challenges in modern bioinformatics. In many living beings (including the human), every gene is present in two copies, inherited from the two parents, the so-called haplotypes. In this paper, we propose a simple combinatorial model for classifying the set of haplotypes in a population according to their responsibility for a certain genetic disease. This model is based on the minimum-ones 2SAT problem with uniform clauses. The minimum-ones 2SAT problem asks for a satisfying assignment to a satisfiable formula in 2CNF which sets a minimum number of variables to true. This problem is well-known to be $\mathcal\{NP\}$-hard, even in the case where all clauses are uniform, i.e., do not contain a positive and a negative literal. We analyze the approximability and present the first non-trivial exact algorithm for the uniform minimum-ones 2SAT problem with a running time of $\mathcal\{O\}$(1.21061n) on a 2SAT formula with n variables. We also show that the problem is fixed-parameter tractable by showing that our algorithm can be adapted to verify in $\mathcal\{O\}^*$(2k) time whether an assignment with at most k true variables exists. },
author = {Böckenhauer, Hans-Joachim, Forišek, Michal, Oravec, Ján, Steffen, Björn, Steinhöfel, Kathleen, Steinová, Monika},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Exact algorithms; fixed-parameter algorithms; minimum-ones 2SAT; haplotypes; exact algorithms},
language = {eng},
month = {10},
number = {3},
pages = {363-377},
publisher = {EDP Sciences},
title = {The Uniform Minimum-Ones 2SAT Problem and its Application to Haplotype Classification},
url = {http://eudml.org/doc/250809},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Böckenhauer, Hans-Joachim
AU - Forišek, Michal
AU - Oravec, Ján
AU - Steffen, Björn
AU - Steinhöfel, Kathleen
AU - Steinová, Monika
TI - The Uniform Minimum-Ones 2SAT Problem and its Application to Haplotype Classification
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/10//
PB - EDP Sciences
VL - 44
IS - 3
SP - 363
EP - 377
AB - Analyzing genomic data for finding those gene variations which are responsible for hereditary diseases is one of the great challenges in modern bioinformatics. In many living beings (including the human), every gene is present in two copies, inherited from the two parents, the so-called haplotypes. In this paper, we propose a simple combinatorial model for classifying the set of haplotypes in a population according to their responsibility for a certain genetic disease. This model is based on the minimum-ones 2SAT problem with uniform clauses. The minimum-ones 2SAT problem asks for a satisfying assignment to a satisfiable formula in 2CNF which sets a minimum number of variables to true. This problem is well-known to be $\mathcal{NP}$-hard, even in the case where all clauses are uniform, i.e., do not contain a positive and a negative literal. We analyze the approximability and present the first non-trivial exact algorithm for the uniform minimum-ones 2SAT problem with a running time of $\mathcal{O}$(1.21061n) on a 2SAT formula with n variables. We also show that the problem is fixed-parameter tractable by showing that our algorithm can be adapted to verify in $\mathcal{O}^*$(2k) time whether an assignment with at most k true variables exists.
LA - eng
KW - Exact algorithms; fixed-parameter algorithms; minimum-ones 2SAT; haplotypes; exact algorithms
UR - http://eudml.org/doc/250809
ER -

References

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