A New Heuristic for Feature Selection by Consistent Biclustering

Given a set of data, biclustering aims at finding simultaneous partitions in biclusters of its samples and of the features which are used for representing the samples. Consistent biclusterings allow to obtain correct classifications of the samples from the known classification of the features, and vice versa, and they are very useful for performing supervised classifications. The problem of finding consistent biclusterings can be seen as a feature selection problem, where the features that are not relevant for classification purposes are removed from the set of data, while the total number of features is maximized in order to preserve information. This feature selection problem can be formulated as a linear fractional 0-1 optimization problem. We propose a reformulation of this problem as a bilevel optimization problem, and we present a heuristic algorithm for an efficient solution of the reformulated problem. Computational experiments show that the presented algorithm is able to find better solutions with respect to the ones obtained by employing previously presented heuristic algorithms

Maximizing the number of solved aircraft conflicts through velocity regulation

International audienceWe propose a model for the maximization of the number of aircraft conflicts that can be solved by performing velocity regulation. The model is mixed-integer as binary variables are used to count solved conflicts and to model alternative choices, while nonlinearities appear in the aircraft separation conditions. The main nonlinearities can however be relaxed by standard reformulations. Numerical results show that the model can be satisfactorily applied at least as a preprocessing step in a conflict avoidance procedure in a given airspace

Using mathematical programming to refine heuristic solutions for network clustering

International audienceWe propose mathematical programming based aproaches to refine graph clustering solutions computed by heuristics. Clustering partitions are refined by applying cluster splitting and a combination of merging and splitting actions. A refinement scheme based on iteratively fixing and releasing integer variables of a mixed-integer quadratic optimization formulation appears to be particularly efficient. Computational experiments show the effectiveness and efficiency of the proposed approaches

Edge ratio and community structure in networks

International audienceA hierarchical divisive algorithm is proposed for identifying communities in complex networks. To that effect, the definition of community in the weak sense of Radicchi et al. _Proc. Natl. Acad. Sci. U.S.A. 101, 2658 _2004__ is extended into a criterion for a bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges. A mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition. This includes an exact solution of the problem of detecting indivisible communities. The resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets. For two problems of the former kind optimal solutions are found; for five problems of the latter kind the edge ratio algorithm always appears to be competitive. Moreover, it provides additional information in several cases, notably through the use of the dendrogram summarizing the resolution. Finally, both algorithms are compared on reduced versions of the data sets of Girvan and Newman _Proc. Natl. Acad. Sci. U.S.A. 99, 7821 _2002__ and of Lancichinetti et al. _Phys. Rev. E 78, 046110 _2008__. Results for these instances appear to be comparable

Range reduction using fixed points

National audienceThe efficiency of sBB depends on many parameters, among which the width of the variable ranges at each node. The fastest range reduction algorithm is called Feasibility-Based Bounds Tightening (FBBT) : it is an iterative procedure that propagates bounds up and down the expression trees [1] representing the constraints in (1), tightening them by using the constraint bounds (-?, 0]. Depending on the instance, and even limited to linear constraints only, FBBT may not converge finitely to its limit point. Tolerance-based termination criteria yield finite termination but, in general, in unbounded time (for every time bound, there is an instance exceeding it). So, although the FBBT is practically fast, its theoretical worst-case complexity status is far from satisfactory

Improving heuristics for network modularity maximization using an exact algorithm

International audienceHeuristics are widely applied to modularity maximization models for the identification of communities in complex networks. We present an approach to be applied as a post-processing to heuristic methods in order to improve their performances. Starting from a given partition, we test with an exact algorithm for bipartitioning if it is worthwhile to split some communities or to merge two of them. A combination of merge and split actions is also performed. Computational experiments show that the proposed approach is effective in improving heuristic results

Locally optimal heuristic for modularity maximization of networks

International audienceCommunity detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics, hybrids, and, in a few papers, exact algorithms. We consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way. This heuristic is compared with the spectral-based hierarchical divisive heuristic of Newman [Proc. Natl. Acad. Sci. USA 103, 8577 (2006).] and with the hierarchical agglomerative heuristic of Clauset, Newman, and Moore [Phys. Rev. E 70, 066111 (2004).]. Computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges. They show that the proposed divisive heuristic gives better results than the divisive heuristic of Newman and than the agglomerative heuristic of Clauset et al

A decomposition-based optimal control approach for aircraft conflict avoidance performed by velocity regulation

International audienceOne of the decisive tasks within the air traffic management is the resolution of aircraft conflict avoidance problems. To avoid conflict, aircraft have to preserve a minimal safety distance between them. In this paper, we present optimal control models and approaches based on speed regulation to perform aircraft conflict avoidance. We consider some aircraft configurations with separable trajectories, i.e., such that trajectories of aircraft pairs exhibit conflict zones which are each other separated in terms of time and/or space. We propose a decomposition of the problem in such a way to solve independently subproblems of the original one

Reformulation of a locally optimal heuristic for modularity maximization

National audienceA network, or graph, G = (V,E) consists of a set of vertices V = {1, . . . , n} and a set of edges E = {1, . . . ,m} connecting vertices. One of the most studied problems in the field of complex systems is to find communities, or clusters, in networks. A community consists of a subset S of the vertices of V where inner edges connecting pairs of vertices of S are more dense than cut edges connecting vertices of S to vertices of V \S. Many criteria have been proposed to evaluate partitions of V into communities

Hierarchical clustering for the identification of communities in networks

National audienceThe analysis of networks and in particular the identification of communities, or clusters, is a topic of active research and attracts an increasing attention in the operations research as well as the physics communities. Complex systems arising in a variety of fields can be represented as networks, or graphs, where the set of vertices is given by the entities under study and the edges represent relations holding for pairs of vertices. A typical example is given by social networks, modeling interactions among people. Other real-life applications include communicatons networks, such as theWorldWide Web, and transportation networks, representing movements of people or goods