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On the frontier of polynomial computations in tropical geometry
 Journal of Symbolic Computation
"... Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We cha ..."
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Abstract. We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving N Phardness and #Phardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions. 1.
Bounds on the number of inference functions of a graphical model
 Formal Power Series and Algebraic Combinatorics (FPSAC 18
, 2006
"... Abstract. Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of ..."
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Abstract. Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of inference functions of any graphical model. This bound is polynomial on the size of the model, for a fixed number of parameters, thus improving the exponential upper bound given by Pachter and Sturmfels [14]. We also show that our bound is tight up to a constant factor, by constructing a family of hidden Markov models whose number of inference functions agrees asymptotically with the upper bound. Finally, we apply this bound to a model for sequence alignment that is used in computational biology.
The minimum evolution problem: Overview and classification
 Networks, In print
, 2008
"... Molecular phylogenetics studies the hierarchical evolutionary relationships among organisms by means of molecular data. These relationships are typically described through a weighted tree, called phylogeny, whose leaves represent the observed organisms, internal vertices represent the intermediate a ..."
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Molecular phylogenetics studies the hierarchical evolutionary relationships among organisms by means of molecular data. These relationships are typically described through a weighted tree, called phylogeny, whose leaves represent the observed organisms, internal vertices represent the intermediate ancestors, and the edges represent evolutionary relationships between pairs of organisms. Molecular phylogenetics provides several criteria to select a phylogeny among plausible alternative ones. Usually, such criteria can be expressed in terms of objective functions, and the phylogenies optimizing them are referred as optimal. One of the most important criteria is Minimum Evolution (ME) which states that the optimal phylogeny for a given set of organisms is the one whose sum of the edge weights is minimal. Finding the phylogeny satisfying the minimum evolution criterion involves the solution of an optimization problem, called Minimum Evolution Problem (MEP), notoriously NPHard. This article offers an overview of the minimum evolution problem, and discusses its specific versions arising from the literature.
An introduction to reconstructing ancestral genomes
 In Proc. Symp. in Applied Mathematics
, 2007
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LatticeBased Minimum Error Rate Training using Weighted FiniteState Transducers with Tropical Polynomial Weights
"... Minimum Error Rate Training (MERT) is a method for training the parameters of a loglinear model. One advantage of this method of training is that it can use the large number of hypotheses encoded in a translation lattice as training data. We demonstrate that the MERT line optimisation can be modelle ..."
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Minimum Error Rate Training (MERT) is a method for training the parameters of a loglinear model. One advantage of this method of training is that it can use the large number of hypotheses encoded in a translation lattice as training data. We demonstrate that the MERT line optimisation can be modelled as computing the shortest distance in a weighted finitestate transducer using a tropical polynomial semiring. 1
Counting Gene Finding Functions
, 2007
"... In biology, a genefinding function is one which takes a DNA sequence and returns the most likely underlying intron/exon structure. Generalizing to graphical models, maps from observed states to their most probable hidden state are called inference functions. While the problem of counting inference ..."
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In biology, a genefinding function is one which takes a DNA sequence and returns the most likely underlying intron/exon structure. Generalizing to graphical models, maps from observed states to their most probable hidden state are called inference functions. While the problem of counting inference functions for a given model is not easy, it has been shown that the number of inference functions grows at most polynomially. In this report, we study the Neyman hidden Markov model of length n, and show that the number of inference functions is exactly 4(n − ⌈n/4⌉) + 2. 1 Genefinding in Biology In all organisms, there is a process where a genomic DNA sequence is transcribed into an RNA sequence, after which the RNA sequence can ultimately go on to become some kind of functional unit. The sequences in the genome that undergo this process are what we refer to as genes, while in between genes in the genome are intergenic sequences that do not code for any functional unit. Very often, the above process also involves the translation of the RNA sequence into an amino acid sequence via a set of rules in which three RNA bases code for