Minimum ratio cover of matrix columns by extreme rays of its induced cone

International audienceGiven a matrix S∈ℝm ×n and a subset of columns R, we study the problem of finding a cover of R with extreme rays of the cone $\mathcal{F}=\{v \in \mathbb{R}^n \mid Sv=\mathbf{0}, v\geq \mathbf{0}\}$, where an extreme ray v covers a column k if vk>0. In order to measure how proportional a cover is, we introduce two different minimization problems, namely the minimum global ratio cover (MGRC) and the minimum local ratio cover (MLRC) problems. In both cases, we apply the notion of the ratio of a vector v, which is given by $\frac{\max_i v_i}{\min_{j\mid v_j > 0} v_j}$. We show that these two problems are NP-hard, even in the case in which |R|=1. We introduce a mixed integer programming formulation for the MGRC problem, which is solvable in polynomial time if all columns should be covered, and introduce a branch-and-cut algorithm for the MLRC problem. Finally, we present computational experiments on data obtained from real metabolic networks

Tropical polar cones, hypergraph transversals, and mean payoff games

We discuss the tropical analogues of several basic questions of convex duality. In particular, the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy. We characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs. We also give a tropical analogue of Farkas lemma, which allows one to check whether a linear inequality is implied by a finite family of linear inequalities. Here, the certificate is a strategy of a mean payoff game. We discuss examples, showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded, and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone.Comment: 27 pages, 6 figures, revised versio

Pseudometrics, The Complex of Ultrametrics, and Iterated Cycle Structures

Every set X, finite of cardinality n say, carries a set M(X) of all possible pseudometrics. It is well known that M(X) forms a convex polyhedral cone whose faces correspond to triangle inequalities. Every point in a convex cone can be expressed as a conical sum of its extreme rays, hence the interest around discovering and classifying such rays. We shall give examples of extreme rays for M(X) exhibiting all integral edge lengths up to half the cardinality of X. By intersecting the cone with the unit cube we obtain the convex polytope of bounded-by-one pseudometrics BM(X). Analogous to extreme rays, every point in a convex polytope arises as a convex combination of extreme points. Extreme rays of BM(X) give rise to very special extreme points of ̄BM(X) as we may normalize a nonzero pseudometric to make its largest distance 1. We shall give a simple and complete characterization of extremeness for metrics with only edge lengths equal to 1/2 and 1. Then we shall use this characterization to give a decomposition result for the upper half of BM(X). BM(X) contains the set of bounded-by-1 pseudoultrametrics, U(X). Ultrametrics satisfy a stronger version of the triangle inequality, and have an interesting structure expressed in terms of partition chains. We will describe the topology of U(X) and its subset of scaled ultrametrics, SU(X), up to homotopy equivalence. Every permutation on a set X can be written as a product of disjoint cycles that cover X. In this way, a permutation generalizes a partition. An iterated cycle structure (ICS) will then be the associated generalization of a partition chain. Analogously, we will compute the “Euler-characteristic” of the set of iterated cycle structures

Constraint-based Analysis of Substructures of Metabolic Networks

Constraint-based methods (CBMs) are promising tools for the analysis of metabolic networks, as they do not require detailed knowledge of the biochemical reactions. Some of these methods only need information about the stoichiometric coefficients of the reactions and their reversibility types, i.e., constraints for steady-state conditions. Nevertheless, CBMs have their own limitations. For example, these methods may be sensitive to missing information in the models. Additionally, they may be slow for the analysis of genome-scale metabolic models. As a result, some studies prefer to consider substructures of networks, instead of complete models. Some other studies have focused on better implementations of the CBMs. In Chapter 2, the sensitivity of flux coupling analysis (FCA) to missing reactions is studied. Genome-scale metabolic reconstructions are comprehensive, yet incomplete, models of real-world metabolic networks. While FCA has proved an appropriate method for analyzing metabolic relationships and for detecting functionally related reactions in such models, little is known about the impact of missing reactions on the accuracy of FCA. Note that having missing reactions is equivalent to deleting reactions, or to deleting columns from the stoichiometric matrix. Based on an alternative characterization of flux coupling relations using elementary flux modes, we study the changes that flux coupling relations may undergo due to missing reactions. In particular, we show that two uncoupled reactions in a metabolic network may be detected as directionally, partially or fully coupled in an incomplete version of the same network. Even a single missing reaction can cause significant changes in flux coupling relations. In case of two consecutive E. coli genome-scale networks, many fully-coupled reaction pairs in the incomplete network become directionally coupled or even uncoupled in the more complete reconstruction. In this context, we found gene expression correlation values being significantly higher for the pairs that remained fully coupled than for the uncoupled or directionally coupled pairs. Our study clearly suggests that FCA results are indeed sensitive to missing reactions. Since the currently available genome-scale metabolic models are incomplete, we advise to use FCA results with care. In Chapter 3, a different, but related problem is considered. Due to the large size of genome-scale metabolic networks, some studies suggest to analyze subsystems, instead of original genome-scale models. Note that analysis of a subsystem is equivalent to deletion of some rows from the stoichiometric matrix, or identically, assuming some internal metabolites to be external. We show mathematically that analysis of a subsystem instead of the original model can lead the flux coupling relations to undergo certain changes. In particular, a pair of (fully, partially or directionally) coupled reactions may be detected as uncoupled in the chosen subsystem. Interestingly, this behavior is the opposite of the flux coupling changes that may happen due to the existence of missing reactions, or equivalently, deletion of reactions. We also show that analysis of organelle subsystems has relatively little influence on the results of FCA, and therefore, many of these subsystems may be studied independent of the rest of the network. In Chapter 4, we introduce a rapid FCA method, which is appropriate for genome-scale networks. Previously, several approaches for FCA have been proposed in the literature, namely flux coupling finder algorithm, FCA based on minimal metabolic behaviors, and FCA based on elementary flux patterns. To the best of our knowledge none of these methods are available as a freely available software. Here, we introduce a new FCA algorithm FFCA (Feasibility-based Flux Coupling Analysis). This method is based on checking the feasibility of a system of linear inequalities. We show on a set of benchmarks that for genome-scale networks FFCA is faster than other existing FCA methods. Using FFCA, flux coupling analysis of genome-scale networks of S. cerevisiae and E. coli can be performed in a few hours on a normal PC. A corresponding software tool is freely available for non-commercial use. In Chapter 5, we introduce a new concept which can be useful in the analysis of fluxes in network substructures. Analysis of elementary modes (EMs) is proven to be a powerful CBM in the study of metabolic networks. However, enumeration of EMs is a hard computational task. Additionally, due to their large numbers, one cannot simply use them as an input for subsequent analyses. One possibility is to restrict the analysis to a subset of interesting reactions, rather than the whole network. However, analysis of an isolated subnetwork can result in finding incorrect EMs, i.e. the ones which are not part of any steady-state flux distribution in the original network. The ideal set of vectors to describe the usage of reactions in a subnetwork would be the set of all EMs projected onto the subset of interesting reactions. Recently, the concept of “elementary flux patterns” (EFPs) has been proposed. Each EFP is a subset of the support (i.e. non-zero elements) of at least one EM. In the present work, we introduce the concept of ProCEMs (Projected Cone Elementary Modes). The ProCEM set can be computed by projecting the flux cone onto the lower-dimensional subspace and enumerating the extreme rays of the projected cone. In contrast to EFPs, ProCEMs are not merely a set of reactions, but from the mathematical point of view they are projected EMs. We additionally prove that the set of EFPs is included in the set of ProCEM supports. Finally, ProCEMs and EFPs are compared in the study of substructures in biological networks

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Cardinality Constrained Optimization Problems

In this thesis, we examine optimization problems with a constraint that allows for only a certain number of variables to be nonzero. This constraint, which is called a cardinality constraint, has received considerable attention in a number of areas such as machine learning, statistics, computational finance, and operations management. Despite their practical needs, most optimization problems with a cardinality constraints are hard to solve due to their nonconvexity. We focus on constructing tight convex relaxations to such problems