An evolutionary approach to solving the maximum size consecutive ones submatrix and related problems

• Main Author: Abo Alsabeh, Rewayda
• Published: University of Essex 2017
• Subjects: QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.732612

The Consecutive Ones Submatrix (C1S) has a vital role in real world applications. Consequently, there are continuous concern and demand to solve this problem via efficient algorithms. These algorithms are judged on the basis of their robustness, ease of use, and their computational time. The main aim of this thesis is to convert a Pure Integer Linear Programming (ILP) with (0, 1)−matrix into Mixed Integer Linear Programming (MILP) by finding the C1S submatrix. Given a (0, 1)−matrix, we consider the C1S problem which aims to maximize the number of columns having only one block of consecutive 1’s in each row by permuting them. We suggest an evolutionary approach to solve the problem. The Genetic Algorithm (GA) is the one proposed here to rearrange the columns of the matrix by pushing them in large blocks of 1’s. We also consider the Consecutive Blocks Minimization (CBM) problem which is related to C1S. A new procedure is proposed to improve the C1S submatrix, which is the column insertion approach. Moreover, preprocessing by minimum degree ordering is also used. On the other hand, we suggest another approach to solve the C1S. It is using the MVEE problem. To pave the way we first solve the problem. Given a set of points C = {x 1 ,x 2 ,...,x m } ⊆ R^n , what is the minimum volume ellipsoid that encloses it? Equally interestingly, one may ask: What is the maximum volume ellipsoid that can be embedded in the set of points without containing any? These problems have a number of applications beside being interesting in their own right. If one requires that at least k of m points, k < m be enclosed in the minimum volume ellipsoid, then the problem becomes more difficult but has the potential, once solved, to detect outliers among the n points. We suggest an evolutionary-type approach for their solution. We will also highlight application areas and include computational results.