Generalized median graphs and applications

• Published in: Journal of combinatorial optimization 2009, Vol.17 (1), p.21-44
• Main Authors: Mukherjee, Lopamudra, Singh, Vikas, Peng, Jiming, Xu, Jinhui, Zeitz, Michael J., Berezney, Ronald
• Format: Article
• Language: English
• Subjects: Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theory of Computation
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-008-9184-7

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.