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**1 - 3**of**3**### Heuristics for the generalized median graph problem

"... Structural approaches for pattern recognition frequently make use of graphs to represent objects. The concept of object similarity is of great importance in pattern recognition. The graph edit distance is often used to measure the similarity between two graphs. It basically consists in the amount o ..."

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Structural approaches for pattern recognition frequently make use of graphs to represent objects. The concept of object similarity is of great importance in pattern recognition. The graph edit distance is often used to measure the similarity between two graphs. It basically consists in the amount of distortion needed to transform one graph into the other. The median graph of a set S of graphs is a graph of S that minimizes the sum of its distances to all other graphs in S. The generalized median graph of S is any graph that minimizes the sum of the distances to all graphs in S. It is the graph that best captures the information contained in S and may be regarded as the best representative of the set. Exact methods for solving the generalized median graph problem are capable to handle only a few number of small graphs. We propose two new heuristics for solving the generalized median graph problem: a greedy adaptive algorithm and a GRASP heuristic. Numerical results indicate that both heuristics can be used to obtain good approximate solutions for the generalized median graph problem, significantly improving the initial solutions and the median graphs. Therefore, the generalized median graph can be effectively computed and used as a better representation than the median graph in a number of relevant pattern recognition applications. This conclusion is supported by experiments with a classification problem, illustrating that the approach based on the generalized median graph is competitive with the k-NN classifier.

### A Bottom-Up implementation of Path-Relinking for Phylogenetic Reconstruction applied to Maximum Parsimony

"... Abstract—In this article we describe a bottom-up implemen-tation of Path-Relinking for Phylogenetic Trees in the context of the resolution of the Maximum Parsimony problem with Fitch optimality criterion. This bottom-up implementation is compared to two versions of an existing top-down implementatio ..."

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Abstract—In this article we describe a bottom-up implemen-tation of Path-Relinking for Phylogenetic Trees in the context of the resolution of the Maximum Parsimony problem with Fitch optimality criterion. This bottom-up implementation is compared to two versions of an existing top-down implementation. We show that our implementation is more efficient, more interesting to compare trees and to give an estimation of the distance between two trees in terms of the number of transformations. Keywords—Phylogenetic Reconstruction, Path-Relinking I.

### GRASP WITH EXTERIOR PATH RELINKING FOR DIFFERENTIAL DISPERSION MINIMIZATION

"... Abstract. We propose several new hybrid heuristics for the differential dis-persion problem are proposed, the best of which consists of a GRASP with sam-pled greedy construction with variable neighborhood search for local improve-ment. The heuristic maintains an elite set of high-quality solutions t ..."

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Abstract. We propose several new hybrid heuristics for the differential dis-persion problem are proposed, the best of which consists of a GRASP with sam-pled greedy construction with variable neighborhood search for local improve-ment. The heuristic maintains an elite set of high-quality solutions throughout the search. After a fixed number of GRASP iterations, exterior path relinking is applied between all pairs of elite set solutions and the best solution found is returned. Exterior path relinking, or path separation, a variant of the more common interior path relinking, is first applied in this paper. In interior path relinking, paths in the neighborhood solution space connecting good solutions are explored between these solutions in the search for improvements. Exte-rior path relinking, as opposed to exploring paths between pairs of solutions, explores paths beyond those solutions. This is accomplished by considering an initiating solution and a guiding solution and introducing in the initiat-ing solution attributes not present in the guiding solution. To complete the process, the roles of initiating and guiding solutions are exchanged. Extensive computational experiments on 190 instances from the literature demonstrate the competitiveness of this algorithm. 1.