The uniform minimum-ones 2SAT problem and its application to haplotype classification
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 3, pp. 363-377.

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 $\mathrm{𝒩𝒫}$-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.

DOI : https://doi.org/10.1051/ita/2010018
Classification : 68Q25,  68R10,  92D20
Mots clés : exact algorithms, fixed-parameter algorithms, minimum-ones 2SAT, haplotypes
@article{ITA_2010__44_3_363_0,
author = {B\"ockenhauer, Hans-Joachim and Fori\v{s}ek, Michal and Oravec, J\'an and Steffen, Bj\"orn and Steinh\"ofel, Kathleen and Steinov\'a, Monika},
title = {The uniform minimum-ones {2SAT} problem and its application to haplotype classification},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {363--377},
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Böckenhauer, Hans-Joachim; Forišek, Michal; Oravec, Ján; Steffen, Björn; Steinhöfel, Kathleen; Steinová, Monika. The uniform minimum-ones 2SAT problem and its application to haplotype classification. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) no. 3, pp. 363-377. doi : 10.1051/ita/2010018. http://www.numdam.org/articles/10.1051/ita/2010018/

[1] B. Aspvall, M.F. Plass and R.E. Tarjan, A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inf. Proc. Lett. 8 (1979) 121-123. | Zbl 0398.68042

[2] H.-J. Böckenhauer and D. Bongartz, Algorithmic Aspects of Bioinformatics. Natural Computing Series, Springer-Verlag (2007). | Zbl 1131.68579

[3] P. Bonizzoni, G.D. Vedova, R. Dondi and J. Li, The haplotyping problem: An overview of computational models and solutions. J. Comput. Sci. Technol. 18 (2003) 675-688. | Zbl 1083.68579

[4] J. Chen, I. Kanj and W. Jia, Vertex cover: further observations and further improvements. J. Algorithms 41 (2001) 280-301. | Zbl 1017.68087

[5] I. Dinur and S. Safra, On the hardness of approximating minimum vertex cover. Ann. Math. 162 (2005) 439-485. | Zbl 1084.68051

[6] D. Gusfield and L. Pitt, A bounded approximation for the minimum cost 2-sat problem. Algorithmica 8 (1992) 103-117. | Zbl 0753.68048

[7] J. Hromkovič, Algorithmics for Hard Problems. Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. Texts in Theoretical Computer Science, An EATCS Series, Springer-Verlag, Berlin (2003). | Zbl 1069.68642

[8] G. Karakostas, A better approximation ratio for the vertex cover problem. Technical Report TR04-084, ECCC (2004). | Zbl 1085.68112

[9] S. Khot and O. Regev, Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci. 74 (2008) 335-349. | Zbl 1133.68061

[10] J. Kiniwa, Approximation of self-stabilizing vertex cover less than 2, in Self Stabilizing Systems (2005) 171-182. | Zbl 1172.68682

[11] E.L. Lawler and D.E. Wood, Branch-and-bound methods: A survey. Operat. Res. 14 (1966) 699-719. | Zbl 0143.42501

[12] J. Li and T. Jiang, A survey on haplotyping algorithms for tightly linked markers. J. Bioinf. Comput. Biol. 6 (2008) 241-259.

[13] J.M. Robson, Finding a maximum independent set in time O(2n/4). Technical Report 1251-01, LaBRI, Université Bordeaux I (2001).

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