1,162 research outputs found

    Algorithms for weighted multidimensional search and perfect phylogeny

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    This dissertation is a collection of papers from two independent areas: convex optimization problems in R[superscript]d and the construction of evolutionary trees;The paper on convex optimization problems in R[superscript]d gives improved algorithms for solving the Lagrangian duals of problems that have both of the following properties. First, in absence of the bad constraints, the problems can be solved in strongly polynomial time by combinatorial algorithms. Second, the number of bad constraints is fixed. As part of our solution to these problems, we extend Cole\u27s circuit simulation approach and develop a weighted version of Megiddo\u27s multidimensional search technique;The papers on evolutionary tree construction deal with the perfect phylogeny problem, where species are specified by a set of characters and each character can occur in a species in one of a fixed number of states. This problem is known to be NP-complete. The dissertation contains the following results on the perfect phylogeny problem: (1) A linear time algorithm when all the characters have two states. (2) A polynomial time algorithm when the number of character states is fixed. (3) A polynomial time algorithm when the number of characters is fixed

    Potential Maximal Clique Algorithms for Perfect Phylogeny Problems

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    Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard combinatorial optimization problems related to minimal triangulations, are broken into subproblems by block subgraphs defined by minimal separators. These ideas were expanded on by Bouchitt\'e and Todinca, who used potential maximal cliques to solve these problems using a dynamic programming approach in time polynomial in the number of minimal separators of a graph. It is known that solutions to the perfect phylogeny problem, maximum compatibility problem, and unique perfect phylogeny problem are characterized by minimal triangulations of the partition intersection graph. In this paper, we show that techniques similar to those proposed by Bouchitt\'e and Todinca can be used to solve the perfect phylogeny problem with missing data, the two- state maximum compatibility problem with missing data, and the unique perfect phylogeny problem with missing data in time polynomial in the number of minimal separators of the partition intersection graph

    Supertree construction by matrix representation with flip

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    A Simple Characterization of the Minimal Obstruction Sets for Three-State Perfect Phylogenies

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    Lam, Gusfield, and Sridhar (2009) showed that a set of three-state characters has a perfect phylogeny if and only if every subset of three characters has a perfect phylogeny. They also gave a complete characterization of the sets of three three-state characters that do not have a perfect phylogeny. However, it is not clear from their characterization how to find a subset of three characters that does not have a perfect phylogeny without testing all triples of characters. In this note, we build upon their result by giving a simple characterization of when a set of three-state characters does not have a perfect phylogeny that can be inferred from testing all pairs of characters

    Improved Lower Bounds on the Compatibility of Multi-State Characters

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    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)f(r) such that, for any set CC of rr-state characters, CC is compatible if and only if every subset of f(r)f(r) characters of CC is compatible. We show that for every r≥2r \ge 2, there exists an incompatible set CC of ⌊r2⌋⋅⌈r2⌉+1\lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1 rr-state characters such that every proper subset of CC is compatible. Thus, f(r)≥⌊r2⌋⋅⌈r2⌉+1f(r) \ge \lfloor\frac{r}{2}\rfloor\cdot\lceil\frac{r}{2}\rceil + 1 for every r≥2r \ge 2. This improves the previous lower bound of f(r)≥rf(r) \ge r given by Meacham (1983), and generalizes the construction showing that f(4)≥5f(4) \ge 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≥4n \ge 4, there exists an incompatible set QQ of ⌊n−22⌋⋅⌈n−22⌉+1\lfloor\frac{n-2}{2}\rfloor\cdot\lceil\frac{n-2}{2}\rceil + 1 quartets over nn labels such that every proper subset of QQ is compatible. We contrast this with a result on the compatibility of triplets: For every n≥3n \ge 3, if RR is an incompatible set of more than n−1n-1 triplets over nn labels, then some proper subset of RR is incompatible. We show this upper bound is tight by exhibiting, for every n≥3n \ge 3, a set of n−1n-1 triplets over nn taxa such that RR is incompatible, but every proper subset of RR is compatible

    Phylogenetic Trees and Their Analysis

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    Determining the best possible evolutionary history, the lowest-cost phylogenetic tree, to fit a given set of taxa and character sequences using maximum parsimony is an active area of research due to its underlying importance in understanding biological processes. As several steps in this process are NP-Hard when using popular, biologically-motivated optimality criteria, significant amounts of resources are dedicated to both both heuristics and to making exact methods more computationally tractable. We examine both phylogenetic data and the structure of the search space in order to suggest methods to reduce the number of possible trees that must be examined to find an exact solution for any given set of taxa and associated character data. Our work on four related problems combines theoretical insight with empirical study to improve searching of the tree space. First, we show that there is a Hamiltonian path through tree space for the most common tree metrics, answering Bryant\u27s Challenge for the minimal such path. We next examine the topology of the search space under various metrics, showing that some metrics have local maxima and minima even with perfect data, while some others do not. We further characterize conditions for which sequences simulated under the Jukes-Cantor model of evolution yield well-behaved search spaces. Next, we reduce the search space needed for an exact solution by splitting the set of characters into mutually-incompatible subsets of compatible characters, building trees based on the perfect phylogenies implied by these sets, and then searching in the neighborhoods of these trees. We validate this work empirically. Finally, we compare two approaches to the generalized tree alignment problem, or GTAP: Sequence alignment followed by tree search vs. Direct Optimization, on both biological and simulated data

    Tracing evolutionary links between species

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    The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe today: fossils, and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM
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