Alexandre S. Freire

Minimum ratio cover of matrix columns by extreme rays of its induced cone

Given a matrix S∈ℝm ×n and a subset of columns R, we study the problem of finding a cover of R with extreme rays of the cone

A dynamic programming algorithm for the tree mapping problem

\mathcal{F}=\{v \in \mathbb{R}^n \mid Sv=\mathbf{0}, v\geq \mathbf{0}\}

A Column Generation Approach for the Graph Matching Problem

, where an extreme ray v covers a column k if vk>0. In order to measure how proportional a cover is, we introduce two different minimization problems, namely the minimum global ratio cover (MGRC) and the minimum local ratio cover (MLRC) problems. In both cases, we apply the notion of the ratio of a vector v, which is given by

Solving the maximum edge biclique packing problem on unbalanced bipartite graphs

\frac{\max_i v_i}{\min_{j\mid v_j > 0} v_j}

An integer linear programming approach for bilinear integer programming

. We show that these two problems are NP-hard, even in the case in which |R|=1. We introduce a mixed integer programming formulation for the MGRC problem, which is solvable in polynomial time if all columns should be covered, and introduce a branch-and-cut algorithm for the MLRC problem. Finally, we present computational experiments on data obtained from real metabolic networks.

A branch-and-bound algorithm for the maximum capture problem with random utilities

Abstract In the graph mapping problem two graphs and a cost function are given as input and the goal is to find a minimum cost mapping of the vertex set of one graph into the vertex set of the other graph. In this paper we introduce a dynamic programming algorithm for the tree mapping problem (i.e., the variant of the problem in which the input graphs are trees), which is still NP-hard, and evaluate its performance with computational experiments.

The biclique k-clustering problem in bipartite graphs and its application in bioinformatics

Graph matching plays a central role in different problems for structural pattern recognition. Examples of applications include matching 3D CAD models, shape matching and medical imaging, to name but a few. In this paper, we present a new integer linear formulation for the problem and employ a combinatorial optimization technique, called “column generation”, in order to solve instances of the problem. We also present computational experiments with generated instances.