CSDL Home IEEE/ACM Transactions on Computational Biology and Bioinformatics 2012 vol.9 Issue No.01 - January/February

Subscribe

Issue No.01 - January/February (2012 vol.9)

pp: 273-285

M. L. Bonet , Dept. of Lenguajes y Sist. lnformaticos (LSI), Univ. Politec. de Catalunya (UPC), Barcelona, Spain

S. Linz , Center for Bioinf. (ZBIT), Univ. of Tubingen, Tubingen, Germany

Katherine St. John , Dept. of Math. & Comput. Sci., City Univ. of New York, Boulevard West, NY, USA

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TCBB.2011.108

ABSTRACT

We show that two important problems that have applications in computational biology are ASP-complete, which implies that, given a solution to a problem, it is NP-complete to decide if another solution exists. We show first that a variation of BETWEENNESS, which is the underlying problem of questions related to radiation hybrid mapping, is ASP-complete. Subsequently, we use that result to show that QUARTET COMPATIBILITY, a fundamental problem in phylogenetics that asks whether a set of quartets can be represented by a parent tree, is also ASP-complete. The latter result shows that Steel's QUARTET CHALLENGE, which asks whether a solution to QUARTET COMPATIBILITY is unique, is coNP-complete.

INDEX TERMS

Phylogeny, DNA, Bioinformatics, Computational biology, Computational complexity, Vegetation,satisfiability., ASP-complete, Betweenness, Quartet Challenge, phylogenetics, Quartet Compatibility

CITATION

M. L. Bonet, S. Linz, Katherine St. John, "The Complexity of Finding Multiple Solutions to Betweenness and Quartet Compatibility",

*IEEE/ACM Transactions on Computational Biology and Bioinformatics*, vol.9, no. 1, pp. 273-285, January/February 2012, doi:10.1109/TCBB.2011.108REFERENCES

- [1] M.E.J. Amaral, J.R. Grant, P.K. Riggs, N.B. Stafuzza, E.A.R. Filho, T. Goldammer, R. Weikard, R.M. Brunner, K.J. Kochan, A.J. Greco, J. Jeong, Z. Cai, G. Lin, A. Prasad, S. Kumar, G.P. Saradhi, B. Mathew, M.A. Kumar, M.N. Miziara, P. Mariani, A.R. Caetano, S.R. Galvão, M.S. Tantia, R.K. Vijh, B. Mishra, S.B. Kumar, V.A. Pelai, A.M. Santana, L.C. Fornitano, B.C. Jones, H. Tonhati, S. Moore, P. Stothard, and J.E. Womack, “A First Generation Whole Genome RH Map of the River Buffalo with Comparison to Domestic Cattle,”
BMC Genomics, vol. 9, article 631, 2008.- [2] A. Ben-Dor, B. Chor, D. Graur, R. Ophir, and D. Pelleg, “From Four-Taxon Trees to Phylogenies: The Case of Mammalian Evolution,”
Proc. Second Ann. Int'l Conf. Computational Molecular Biology (RECOMB '98), pp. 9-19, 1998.- [3] V. Berry, T. Jiang, P. Kearney, M. Li, and T. Wareham, “Quartet Cleaning: Improved Algorithms and Simulations,”
Proc. Seventh Ann. European Symp. Algorithms (ESA '99), pp. 313-324, 1999.- [4] V. Berry and O. Gascuel, “Inferring Evolutionary Trees with Strong Combinatorial Evidence,”
Theoretical Computer Science, vol. 240, no. 2, pp. 271-298, 2000.- [5] S. Böcker, D. Bryant, A.W.M. Dress, and M. Steel, “Algorithmic Aspects of Tree Amalgamation,”
J. Algorithms, vol. 37, pp. 522-537, 2000.- [6] D. Bryant and M. Steel, “Extension Operations on Sets of Leaf-Labelled Trees,”
Advances in Applied Math., vol. 16, no. 4, pp. 425-453, 1995.- [7] P. Buneman, “A Note on the Metric Properties of Trees,”
J. Combinatorial Theory Series B, vol. 17, pp. 48-50, 1974.- [8] B. Chor and M. Sudan, “A Geometric Approach to Betweenness,”
SIAM J. Discrete Math., vol. 11, no. 4, pp. 511-523, 1998.- [9] B.P. Chowdhary, T. Raudsepp, S.R. Kata, G. Goh, L.V. Millon, V. Allan, F. Piumi, G. Guerin, J. Swinburne, M. Binns, T.L. Lear, J. Mickelson, J. Murray, D.F. Antczak, J.E. Womack, and L.C. Skow, “The First-Generation Whole-Genome Radiation Hybrid Map in the Horse Identifies Conserved Segments in Human and Mouse Genomes,”
Genome Research, vol. 13, nos. 742-751, 2003.- [10] S.A. Cook, “The Complexity of Theorem-Proving Procedures,”
Proc. the Third Ann. ACM Symp. Theory of Computing, pp. 151-158, 1971.- [11] D. Cox, M. Burmeister, E. Price, S. Kim, and R. Myers, “Radiation Hybrid Mapping: A Somatic Cell Genetic Method for Constructing High-Resolution Maps of Mammalian Chromosomes,”
Science, vol. 250, no. 4978, pp. 245-250, 1990.- [12] C. Dowden, “On the Maximum Size of Minimal Definitive Quartet Sets,”
Discrete Math., vol. 310, pp. 2546-2549, 2010.- [13] P. Erdös, M. Steel, L. Székély, and T. Warnow, “A Few Logs Suffice to Build (Almost) All Trees: Part II,”
Theoretical Computer Science, vol. 221, pp. 77-118, 1999.- [14] L.R. Foulds and R.L. Graham, “The Steiner Problem in Phylogeny Is NP-Complete,”
Advances in Applied Math., vol. 3, no. 1, pp. 43-49, 1982.- [15] M. Garey and D. Johnson,
Computers and Intractability: A Guide to the Theory of NP-Completeness. A Series of Books in the Mathematical Sciences. WH Freeman and Company, 1979.- [16] J. Gramm, A. Nickelsen, and T. Tantau, “Fixed-parameter Algorithms in Phylogenetics,”
The Computer J., vol. 51, pp. 79-101, 2007.- [17] S. Grünewald, P. Humphries, and C. Semple, “Quartet Compatibility and the Quartet Graph,”
The Electronic J. Combinatorics, vol. 15, no. R103, 2008.- [18] G. Gyapay, K. Schmitt, C. Fizames, H. Jones, N. Vega-Czarny, D. Spillett, D. Muselet, J.-F. Prud'Homme, C. Dib, C. Auffray, J. Morissette, J. Weissenbach, and P.N. Goodfellow, “A Radiation Hybrid Map of the Human Genome,”
Human Molecular Genetics, vol. 5, no. 3, pp. 339-346, 1996.- [19] M. Habib and J. Stacho, “Unique Perfect Phylogeny Is NP-Hard,”
Combinatorial Pattern Matching (CPM 2011), LNCS, vol. 6661, pp. 132-146, 2011.- [20] D. Huson, S. Nettles, and T. Warnow, “Disk-Covering, a Fast-Converging Method for Phylogenetic Tree Reconstruction,”
Computational Biology, vol. 6, pp. 369-386, 1999.- [21] T. Jiang, P. Kearney, and M. Li, “Some Open Problems in Computational Molecular Biology,”
J. Algorithms, vol. 34, no. 1, pp. 194-201, 2000.- [22] L. Juban, “Dichotomy Theorem for the Generalized Unique Satisfiability Problem,”
Proc. 12th Int'l Symp. Fundamentals of Computation Theory (FCT '99), pp. 327-338, 1999.- [23] J. Opatrny, “Total Ordering Problem,”
SIAM J. Computing, vol. 8, no. 1, pp. 111-114, 1979.- [24] C. Papadimitriou,
Computational Complexity. John Wiley and Sons Ltd., 2003.- [25] S. Roch, “A Short Proof that Phylogenetic Tree Reconstruction by Maximum Likelihood Is Hard,”
IEEE/ACM Trans. Computational Biology and Bioinformatics, vol. 3, no. 1, pp. 92-94, Jan.-Mar. 2006.- [26] C. Semple and M. Steel, “A Characterization for a Set of Partial Partitions to Define an X-tree,”
Discrete Math., vol. 247, no. 1, pp. 169-186, 2002.- [27] C. Semple and M. Steel,
Phylogenetics. Oxford Univ. Press, 2003.- [28] M. Steel, “The Complexity of Reconstructing Trees from Qualitative Characters and Subtrees,”
J. Classification, vol. 9, pp. 91-116, 1992.- [29] M. Steel, http://www.math.canterbury.ac.nz/~m.steel challen ges.shtml#Hundred, 2011.
- [30] G. Wu, J. You, and G. Lin, “A Lookahead Branch-and-Bound Algorithm for the Maximum Quartet Consistency Problem,”
Proc. Workshop Algorithms in Bioinformatics (WABI '05), pp. 65-76, 2005.- [31] T. Yato and T. Seta, “Complexity and Completeness of Finding Another Solution and Its Application to Puzzles,”
IEICE Trans. Fundamentals of Electronics, Comm. and Computer Sciences, vol. E86-A, no. 5, pp. 1052-1060, 2003. |