Abstract
We study a long standing conjecture on the necessary and sufficient conditions for the compatibility of multi-state characters: There exists a function f(r) such that, for any set C of r-state characters, C is compatible if and only if every subset of f(r) characters of C is compatible. We show that for every r ≥ 2, there exists an incompatible set C of \(\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1\) r-state characters such that every proper subset of C is compatible. Thus, \(f(r) \ge \lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1\) for every r ≥ 2. This improves the previous lower bound of f(r) ≥ r given by Meacham (1983), and generalizes the construction showing that f(4) ≥ 5 given by Habib and To (2011). We prove our result via a result on quartet compatibility that may be of independent interest: For every integer n ≥ 4, there exists an incompatible set Q of \(\lfloor\frac{n-2}{2}\rfloor\cdot\lceil\frac{n-2}{2}\rceil + 1\) quartets over n labels such that every proper subset of Q is compatible. We contrast this with a result on the compatibility of triplets: For every n ≥ 3, if R is an incompatible set of more than n − 1 triplets over n labels, then some proper subset of R is incompatible. We show this bound is tight by exhibiting, for every n ≥ 3, a set of n − 1 triplets over n taxa such that R is incompatible, but every proper subset of R is compatible.
This work was supported in part by the National Science Foundation under grants CCF-1017189 and DEB-0829674.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwala, R., Fernández-Baca, D.: A polynomial-time algorithm for the perfect phylogeny problem when the number of character states is fixed. SIAM Journal on Computing 23(6), 1216–1224 (1994)
Aho, A.V., Sagiv, Y., Szymanski, T.G., Ullman, J.D.: Inferring a tree from lowest common ancestors with an application to the optimization of relational expressions. SIAM Journal on Computing 10(3), 405–421 (1981)
Bodlaender, H., Fellows, M., Warnow, T.: Two Strikes against Perfect Phylogeny. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 273–283. Springer, Heidelberg (1992)
Bryant, D., Steel, M.: Extension operations on sets of leaf-labelled trees. Advances in Applied Mathematics 16, 425–453 (1995)
Buneman, P.: The recovery of trees from measurements of dissimilarity. In: Mathematics in the Archeological and Historical Sciences, pp. 387–395. Edinburgh University Press (1971)
Colonius, H., Schulze, H.H.: Tree structures for proximity data. British Journal of Mathematical and Statistical Psychology 34(2), 167–180 (1981)
Dekker, M.C.H.: Reconstruction Methods for Derivation Trees. Master’s thesis, Vrije Universiteit, Amsterdam, Netherlands (1986)
Dietrich, M., McCartin, C., Semple, C.: Bounding the maximum size of a minimal definitive set of quartets. Information Processing Letters 112(16), 651–655 (2012)
Dress, A., Steel, M.: Convex tree realizations of partitions. Applied Mathematics Letters 5(3), 3–6 (1992)
Estabrook, G.F., Johnson, J., McMorris, F.R.: A mathematical foundation for the analysis of cladistic character compatibility. Mathematical Biosciences 29(1-2), 181–187 (1976)
Fernández-Baca, D.: The Perfect Phylogeny Problem. In: Steiner Trees in Industry, pp. 203–234. Kluwer (2001)
Fitch, W.M.: Toward finding the tree of maximum parsimony. In: Proceedings of the 8th International Conference on Numerical Taxonomy, pp. 189–230 (1975)
Fitch, W.M.: On the problem of discovering the most parsimonious tree. The American Naturalist 111(978), 223–257 (1977)
Grünewald, S., Huber, K.T.: Identifying and defining trees. In: Gascuel, O., Steel, M. (eds.) Reconstructing Evolution: New Mathematical and Computational Advances. Oxford University Press (2007)
Gusfield, D.: Efficient algorithms for inferring evolutionary trees. Networks 21(1), 19–28 (1991)
Habib, M., To, T.H.: On a Conjecture about Compatibility of Multi-states Characters. In: Przytycka, T.M., Sagot, M.-F. (eds.) WABI 2011. LNCS, vol. 6833, pp. 116–127. Springer, Heidelberg (2011)
Kannan, S., Warnow, T.: Inferring Evolutionary History From DNA Sequences. SIAM Journal on Computing 23(4), 713–737 (1994)
Kannan, S., Warnow, T.: A fast algorithm for the computation and enumeration of perfect phylogenies. SIAM Journal on Computing 26(6), 1749–1763 (1997)
Lam, F., Gusfield, D., Sridhar, S.: Generalizing the Splits Equivalence Theorem and Four Gamete Condition: Perfect Phylogeny on Three-State Characters. SIAM Journal on Discrete Mathematics 25(3), 1144–1175 (2011)
Meacham, C.A.: Theoretical and computational considerations of the compatibility of qualitative taxonomic characters. In: Numerical Taxonomy. Nato ASI Series, vol. G1, Springer (1983)
Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press (2003)
Shutters, B., Fernández-Baca, D.: A simple characterization of the minimal obstruction sets for three-state perfect phylogenies. Applied Mathematics Letters 25(9), 1226–1229 (2012)
Steel, M.: Personal communications (2012)
Steel, M.: The complexity of reconstructing trees from qualitative characters and subtrees. Journal of Classification 9(1), 91–116 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shutters, B., Vakati, S., Fernández-Baca, D. (2012). Improved Lower Bounds on the Compatibility of Quartets, Triplets, and Multi-state Characters. In: Raphael, B., Tang, J. (eds) Algorithms in Bioinformatics. WABI 2012. Lecture Notes in Computer Science(), vol 7534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33122-0_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-33122-0_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33121-3
Online ISBN: 978-3-642-33122-0
eBook Packages: Computer ScienceComputer Science (R0)