Javier Yáñez

The robust coloring problem

Abstract Some problems can be modeled as graph coloring ones for which the criterion of minimizing the number of used colors is replaced by another criterion maintaining the number of colors as a constraint. Some examples of these problem types are introduced; it would be the case, for instance, of the problem of scheduling the courses at a university with a fixed number of time slots––the colors––and with the objective of minimizing the probability to include an edge to the graph with its endpoints equally colored. Based on this example, the new coloring problem introduced in this paper will be denoted as the Robust coloring problem, RCP for short. It is proved that this optimization problem is NP-hard and, consequently, only small-size problems could be solved with exact algorithms based on mathematical programming models; otherwise, for large size problems, some heuristics are needed in order to obtain appropriate solutions. A genetic algorithm which solves the RCP is outlined.

A coloring fuzzy graph approach for image classification

One of the main problems in practice is the difficulty in dealing with membership functions. Many decision makers ask for a graphical representation to help them to visualize results. In this paper, we point out that some useful tools for fuzzy classification can be derived from fuzzy coloring procedures. In particular, we bring here a crisp grey coloring algorithm based upon a sequential application of a basic black and white binary coloring procedure, already introduced in a previous paper [D. Gomez, J. Montero, J. Yanez, C. Poidomani, A graph coloring algorithm approach for image segmentation, Omega, in press]. In this article, the image is conceived as a fuzzy graph defined on the set of pixels where fuzzy edges represent the distance between pixels. In this way, we can obtain a more flexible hierarchical structure of colors, which in turn should give useful hints about those classes with unclear boundaries.

Parameter setting of the Hopfield network applied to TSP

The major drawbacks of the continuous Hopfield network (CHN) model when it is used to solve some combinatorial problems, for instance, the traveling salesman problem (TSP), are the non feasibility of the obtained solutions and the trial-and-error setting values process of the model parameters. In this paper, both drawbacks are avoided by introducing a set of analytical conditions guaranteeing that any equilibrium point of the CHN characterizes a tour for the TSP. In this way, any instance of the TSP can be solved with this parameter setting. Some computational experiences are also included, allowing the solution of instances with sizes of up to 1000 cities.

Soft dimension theory

Classical dimension theory, when applied to preference modeling, is based upon the assumption that linear ordering is the only elemental notion for rationality. In fact, crisp preferences are in some way decomposed into basic criteria, each one being a linear order. In this paper, we propose that indeed dimension is relative to a previous idea of rationality, but such a rationality is not unique. In particular, we explore alternative approaches to dimension, based upon a more general representation and allowing different classes of orders for basic criteria. In this way, classical dimension theory is generalized. As a first consequence, we explore the existence of crisp preference representations not being based upon linear orders. As a second consequence, it is suggested that an analysis of valued preference relations can be developed in terms of the representations of all α-cuts.

A Poset Dimension Algorithm

This article presents an algorithm which computes the dimension of an arbitrary finite poset (partial order set). This algorithm is based on the chromatic number of a graph instead of the classical approach based on the chromatic number of some hypergraph. The relation between both approaches is analyzed. With this algorithm, the dimension of many modest size posets can be computed. Otherwise, an upper bound for the poset dimension is obtained. Some computational results are included.

The graph coloring problem: A neuronal network approach ☆

Solution of an optimization problem with linear constraints through the continuous Hopfield network (CHN) is based on an energy or Lyapunov function that decreases as the system evolves until a local minimum value is attained. This approach is extended in to optimization problems with quadratic constraints. As a particular case, the graph coloring problem (GCP) is analyzed. The mapping procedure and an appropriate parameter-setting procedure are detailed. To test the theoretical results, some computational experiments solving the GCP are shown.

A Divide-and-Link algorithm for hierarchical clustering in networks

This paper introduces a hierarchical clustering algorithm in networks based upon a first divisive stage to break the graph and a second linking stage which is used to join nodes. As a particular case, this algorithm is applied to the specific problem of community detection in social networks, where a betweenness measure is considered for the divisive criterion and a similarity measure associated to data is used for the linking criterion. We show that this algorithm is very flexible as well as quite competitive (from both a performance and a computational complexity point of view) in relation with a set of state-of-the-art algorithms. Furthermore, the output given by the proposed algorithm allows to show in a dynamic and interpretable way the evolution of how the groups are split in the network.

A continuous Hopfield network equilibrium points algorithm

The continuous Hopfield network (CHN) is a classical neural network model. It can be used to solve some classification and optimization problems in the sense that the equilibrium points of a differential equation system associated to the CHN is the solution to those problems. The Euler method is the most widespread algorithm to obtain these CHN equilibrium points, since it is the simplest and quickest method to simulate complex differential equation systems. However, this method is highly sensitive with respect to initial conditions and it requires a lot of CPU time for medium or greater size CHN instances. In order to avoid these shortcomings, a new algorithm which obtains one equilibrium point for the CHN is introduced in this paper. It is a variable time-step method with the property that the convergence time is shortened; moreover, its robustness with respect to initial conditions will be proven and some computational experiences will be shown in order to compare it with the Euler method.

Searching for the dimension of valued preference relations

The more information a preference structure gives, the more sophisticated representation techniques are necessary, so decision makers can have a global view of data and therefore a comprehensive understanding of the problem they are faced with. In this paper we propose to explore valued preference relations by means of a search for the number of underlying criteria allowing its representation in real space. A general representation theorem for arbitrary crisp binary relations is obtained, showing the difference in representation between incomparability-related to the intersection operator-and other inconsistencies-related to the union operator. A new concept of dimension is therefore proposed, taking into account inconsistencies in source of information. Such a result is then applied to each alpha-cut of valued preference relations

Structural properties of continuum systems

Abstract This paper deals with monotonic systems where the performance levels of the system and its components range from perfect functioning to complete failure, allowing any intermediate state in the unit interval. In particular, general bounds are found for the reliability of such systems. Moreover, it is shown how these continuum systems can be approached by finite multistate systems.