Fast recognition of classes of almost-median graphs

Journal of Combinatorial Optimization, 2009

We study the so-called Generalized Median graph problem where the task is to construct a prototype (i.e., a ‘model’) from an input set of graphs. While our primary motivation comes from an important biological imaging application, the problem effectively captures many vision (e.g., object recognition) and learning problems, where graphs are increasingly being adopted as a powerful representation tool. Existing techniques for his problem are evolutionary search based; in this paper, we propose a polynomial time algorithm based on a linear programming formulation. We propose an additional algorithm based on a bi-level method to obtain solutions arbitrarily close to the optimal in (worst case) non-polynomial time. Within this new framework, one can optimize edit distance functions that capture similarity by considering vertex labels as well as he graph structure simultaneously. We first discuss experimental evaluations in context of molecular image analysis problems—he methods will provide the basis for building a topological map of all 23 pairs of the human chromosome. Later, we include (a) applications to other biomedical problems and (b) evaluations on a public pattern recognition graph database.

2007

We study the so-called Generalized Median graph problem where the task is to to construct a prototype (i.e., a 'model') from an input set of graphs. The problem finds applications in many vision (e.g., object recognition) and learning problems where graphs are increasingly being adopted as a representation tool. Existing techniques for this problem are evolutionary search based; in this paper, we propose a polynomial time algorithm based on a linear programming formulation. We present an additional bi-level method to obtain solutions arbitrarily close to the optimal in non-polynomial time (in worst case). Within this new framework, one can optimize edit distance functions that capture similarity by considering vertex labels as well as the graph structure simultaneously. In context of our motivating application, we discuss experiments on molecular image analysis problems - the methods will provide the basis for building a topological map of all pairs of the human chromosome.

Algorithmica, 2010

The median (antimedian) set of a profile π = (u 1 , . . . , u k ) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness i d(x, u i ). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes.

SIAM Journal on Discrete Mathematics, 1999

Let M (m, n) be the complexity of checking whether a graph G with m edges and n vertices is a median graph. We show that the complexity of checking whether G is triangle-free is at most M (m, m). Conversely, we prove that the complexity of checking whether a given graph is a median graph is at most O(m log n) + T (m log n, n), where T (m, n) is the complexity of finding all triangles of the graph. We also demonstrate that, intuitively speaking, there are as many median graphs as there are triangle-free graphs. Finally, these results enable us to prove that the complexity of recognizing planar median graphs is linear.

Median graph is an important new concept introduced to represent a set of graphs by a representative graph. Computing the median graph is an NP- Complete problem. In this paper, we propose an approximate algorithm for computing the median graph. Our algorithm performs in two steps. It first carries out a node reduction process using a clustering method to extract a sub- set of most representative node labels. It then searches for the median graph candidates from the reduced subset of node labels according to a deterministic strategy to explore the candidate space. Comparison with the genetic search based algorithm will be reported. This algorithm can be used to build a graph clustering algorithm.

Pattern Recognition, 2010

The median graph has been presented as a useful tool to represent a set of graphs. Nevertheless its computation is very complex and the existing algorithms are restricted to use limited amount of data. In this paper we propose a new approach for the computation of the median graph based on graph embedding. Graphs are embedded into a vector space and the median is computed in the vector domain. We have designed a procedure based on the weighted mean of a pair of graphs to go from the vector domain back to the graph domain in order to obtain a final approximation of the median graph. Experiments on three different databases containing large graphs show that we succeed to compute good approximations of the median graph. We have also applied the median graph to perform some basic classification tasks achieving reasonable good results. These experiments on real data open the door to the application of the median graph to a number of more complex machine learning algorithms where a representative of a set of graphs is needed.

European Journal of Combinatorics, 2000

Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasi-median graphs as well as pseudo-median graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5-wheels, induced subgraphs of hyperoctahedra (alias cocktail party graphs), and 2-connected bridged graphs not containing K 4 or K 1,1,3 as an induced subgraph. As a consequence one obtains that every finite weakly median graph is l 1-embeddable, that is, it embeds as a metric subspace into some R n equipped with the 1-norm.

2008 19th International Conference on Pattern Recognition, 2008

Graphs are powerful data structures that have many attractive properties for object representation. However, some basic operations are difficult to define and implement, for instance, how to obtain a representative of a set of graphs. The median graph has been defined for that purpose, but existing algorithms are computationally complex and have a very limited applicability. In this paper we propose a new approach for the computation of the median graph based on graph embedding in vector spaces. Experiments on a real database containing large graphs show that we succeed to compute good approximations of the median graph. We have also applied the median graph to perform some basic classification tasks achieving reasonable good results.

Networks, 2010

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r ≥ 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H, respectively, and that dH (G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H.

A periphery transversal of a median graph G is introduced as a set of vertices that meets all the peripheral subgraphs of G. Using this concept, median graphs with geodetic number 2 are characterized in two ways. They are precisely the median graphs that contain a periphery transversal of order 2 as well as the median graphs for which there exists a profile such that the remoteness function is constant on G. Moreover, an algorithm is presented that decides in O(m log n) time whether a given graph G with n vertices and m edges is a median graph with geodetic number 2. Several additional structural properties of the remoteness function on hypercubes and median graphs are obtained and some problems listed.

European Journal of Operational Research, 2016

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 a 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 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 and comparisons with algorithm k-NN.

Computer Vision and Image Understanding, 2011

The median graph has been shown to be a good choice to obtain a representative of a set of graphs. However, its computation is a complex problem. Recently, graph embedding into vector spaces has been proposed to obtain approximations of the median graph. The problem with such an approach is how to go from a point in the vector space back to a graph in the graph space. The main contribution of this paper is the generalization of this previous method, proposing a generic recursive procedure that permits to recover the graph corresponding to a point in the vector space, introducing only the amount of approximation inherent to the use of graph matching algorithms. In order to evaluate the proposed method, we compare it with the set median and with the other state-of-the-art embedding-based methods for the median graph computation. The experiments are carried out using four different databases (one semi-artificial and three containing real-world data). Results show that with the proposed approach we can obtain better medians, in terms of the sum of distances to the training graphs, than with the previous existing methods.► Approximate median graph computation. ► Graph embedding into vector spaces. ► Recursive algorithms. ► Graph matching. ► Structural pattern recognition.

Algorithmica, 2020

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes label the vertices of a graph in a such a way that given the labels of a source node and a destination node, it is possible to compute efficiently the port number of the edge from the source that heads in the direction of the destination. One of important problems is finding natural classes of graphs admitting distance and/or routing labeling schemes with labels of polylogarithmic size. In this paper, we show that the class of cube-free median graphs on n nodes enjoys distance and routing labeling schemes with labels of O(log 3 n) bits.

Discussiones Mathematicae Graph Theory, 2010

The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join

2010

In this paper we consider two existing methods to generate a representative of a given set of graphs, that satisfy the following two conditions. On the one hand, that they are applicable to graphs with any kind of labels in nodes and edges and on the other hand, that they can handle relatively large amount of data. Namely, the approximated algorithms to compute the Median Graph via graph embedding and a new method to compute the Barycenter Graph. Our contribution is to give a new algorithm for the barycenter computation and to compare it to the median Graph. To compare these two representatives, we take into account algorithmic considerations and experimental results on the quality of the representative and its robustness, on several datasets.