Fábio Viduani Martinez

Algorithms for terminal Steiner trees

The terminal Steiner tree problem (TST) consists of finding a minimum cost Steiner tree where each terminal is a leaf. We describe a factor 2@r-@r/(3@r-2) approximation algorithm for the TST, where @r is the approximation factor of a given algorithm for the Steiner tree problem. Considering the current best value of @r, this improves a previous 3.10 factor to 2.52. For the TST restricted to instances where all edge costs are either 1 or 2, we improve the approximation factor from 1.60 to 1.42.

Enumerating Precursor Sets of Target Metabolites in a Metabolic Network

We present the first exact method based on the topology of a metabolic network to find minimal sets of metabolites (called precursors) sufficient to produce a set of target metabolites. In contrast with previous proposals, our model takes into account self-regenerating metabolites involved in cycles, which may be used to generate target metabolites from potential precursors. We analyse the complexity of the problem and we propose an algorithm to enumerate all minimal precursor sets for a set of target metabolites. The algorithm can be applied to identify a minimal medium necessary for a cell to ensure some metabolic functions. It can be used also to check inconsistencies caused by misannotations in a metabolic network. We present two illustrations of these applications.

Repetition-free longest common subsequence

We study the following problem. Given two sequences x and y over a finite alphabet, find a repetition-free longest common subsequence of x and y. We show several algorithmic results, a computational complexity result, and we describe a preliminary experimental study based on the proposed algorithms. We also show that this problem is APX-hard.

On the family-free DCJ distance and similarity

Structural variation in genomes can be revealed by many (dis)similarity measures. Rearrangement operations, such as the so called double-cut-and-join (DCJ), are large-scale mutations that can create complex changes and produce such variations in genomes. A basic task in comparative genomics is to find the rearrangement distance between two given genomes, i.e., the minimum number of rearragement operations that transform one given genome into another one. In a family-based setting, genes are grouped into gene families and efficient algorithms have already been presented to compute the DCJ distance between two given genomes. In this work we propose the problem of computing the DCJ distance of two given genomes without prior gene family assignment, directly using the pairwise similarities between genes. We prove that this new family-free DCJ distance problem is APX-hard and provide an integer linear program to its solution. We also study a family-free DCJ similarity and prove that its computation is NP-hard.

Algorithms for terminal steiner trees

The terminal Steiner tree problem (TST) consists of finding a minimum cost Steiner tree where each terminal is a leaf. We describe a factor 2ρ – ρ/(3ρ–2) approximation algorithm for the TST, where ρ is the approximation factor of a given algorithm for the Steiner tree problem. Considering the current best value of ρ, this improves a previous 3.10 factor to 2.52. For the TST restricted to instances where all edge costs are either 1 or 2, we improve the approximation factor from 1.60 to 1.42.

Approximating the DCJ distance of balanced genomes in linear time

BackgroundRearrangements are large-scale mutations in genomes, responsible for complex changes and structural variations. Most rearrangements that modify the organization of a genome can be represented by the double cut and join (DCJ) operation. Given two balanced genomes, i.e., two genomes that have exactly the same number of occurrences of each gene in each genome, we are interested in the problem of computing the rearrangement distance between them, i.e., finding the minimum number of DCJ operations that transform one genome into the other. This problem is known to be NP-hard.ResultsWe propose a linear time approximation algorithm with approximation factor O(k) for the DCJ distance problem, where k is the maximum number of occurrences of any gene in the input genomes. Our algorithm works for linear and circular unichromosomal balanced genomes and uses as an intermediate step an O(k)-approximation for the minimum common string partition problem, which is closely related to the DCJ distance problem.ConclusionsExperiments on simulated data sets show that our approximation algorithm is very competitive both in efficiency and in quality of the solutions.

On the Multichromosomal Hultman Number

The number of cycles of a breakpoint graph is one of the notable parameters to solve distance problems in comparative genomics. For a fixed c, the number of linear unichromosomal genomes with n genes such that the breakpoint graph has c disjoint cycles, the Hultman number, is already determined. In this work we extend this result to multichromosomal genomes, providing formulas to compute the number of multichromosal genomes having a fixed number of cycles and/or paths.

On the Family-Free DCJ Distance

Structural variation in genomes can be revealed by many (dis)similarity measures. Rearrangement operations, such as the so called double-cut-and-join (DCJ), are large-scale mutations that can create complex changes and produce such variations in genomes. A basic task in comparative genomics is to find the rearrangement distance between two given genomes, i.e., the minimum number of rearragement operations that transform one given genome into another one. In a family-based setting, genes are grouped into gene families and efficient algorithms were already proposed to compute the DCJ distance between two given genomes. In this work we propose the problem of computing the DCJ distance of two given genomes without prior gene family assignment, directly using the pairwise similarity between genes. We propose a new family-free DCJ distance, prove that the family-free DCJ distance problem is APX-hard, and provide an integer linear program to its solution.

Algorithms for Computing the Family-Free Genomic Similarity Under DCJ

The genomic similarity is a large-scale measure for comparing two given genomes. In this work we study the (NP-hard) problem of computing the genomic similarity under the DCJ model in a setting that does not assume that the genes of the compared genomes are grouped into gene families. This problem is called family-free DCJ similarity. Here we propose an exact ILP algorithm to solve it, we show its APX-hardness, and we present three combinatorial heuristics, with computational experiments comparing their results to the ILP. Experiments on simulated datasets show that the proposed heuristics are very fast and even competitive with respect to the ILP algorithm for some instances.

A Linear Time Approximation Algorithm for the DCJ Distance for Genomes with Bounded Number of Duplicates

Rearrangements are large-scale mutations in genomes, responsible for complex changes and structural variations. Most rearrangements that modify the organization of a genome can be represented by the double cut and join (DCJ) operation. Given two genomes with the same content, so that we have exactly the same number of copies of each gene in each genome, we are interested in the problem of computing the rearrangement distance between them, i.e., finding the minimum number of DCJ operations that transform one genome into the other. We propose a linear time approximation algorithm with approximation factor O(k) for the DCJ distance problem, where k is the maximum number of duplicates of any gene in the input genomes. Our algorithm uses as an intermediate step an O(k)-approximation for the minimum common string partition problem, which is closely related to the DCJ distance problem. Experiments on simulated data sets show that the algorithm is very competitive both in efficiency and quality of the solutions.