No abstract

No abstract

In an incremental optimization problem, we are given a feasible solution x0 of an optimization problem P, and we want to make an incremental change in x0 that will result in the greatest improvement in the objective function. In this paper, we study the incremental optimization versions of six well-known network problems. We present a strongly polynomial algorithm for the incremental minimum spanning tree problem. We show that the incremental minimum cost flow problem and the incremental maximum flow problem can be solved in polynomial time using Lagrangian relaxation. We consider two versions of the incremental minimum shortest path problem, where increments are measured via arc inclusions and arc exclusions. We present a strongly polynomial time solution for the arc inclusion version and show that the arc exclusion version is NP-complete. We show that the incremental minimum cut problem is NP-complete and that the incremental minimum assignment problem reduces to the minimum exact matching problem, for which a randomized polynomial time algorithm is known.

Binary classification refers to supervised techniques that split a set of points in two classes, with respect to a training set of points whose membership is known for each class. Binary classification plays a central role in the solution of many scientific, financial, engineering, medical and biological problems. Many methods with good classification accuracy are currently available. This work shows how a binary classification problem can be expressed in terms of a generalized eigenvalue problem. A new regularization technique is proposed, which gives results that are comparable to other techniques in use, in terms of classification accuracy. The advantage of this method relies in its lower computational complexity with respect to the existing techniques based on generalized eigenvalue problems. Finally, the method is compared with other methods using benchmark data sets.