A Metaheuristic Approach to the C1S Problem

Rewayda Abo-Alsabeh,Abdellah Salhi

doi:10.24996/ijs.2021.62.1.20

Rewayda Abo-Alsabeh, Abdellah Salhi

Open Access

https://doi.org/10.24996/ijs.2021.62.1.20

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Abstract

Given a binary matrix, finding the maximum set of columns such that the resulting submatrix has the Consecutive Ones Property (C1P) is called the Consecutive Ones Submatrix (C1S) problem. There are solution approaches for it, but there is also a room for improvement. Moreover, most of the studies of the problem use exact solution methods. We propose an evolutionary approach to solve the problem. We also suggest a related problem to C1S, which is the Consecutive Blocks Minimization (CBM). The algorithm is then performed on real-world and randomly generated matrices of the set covering type.

Highlights

The problem of consecutive ones submatrix on a binary matrix has been known since the 1950s
Input a binary matrix A; Input the rate of crossover and the rate of mutation; Generate a random population of permutations of columns of matrix A; Evaluate the fitness of the permutations according to some fitness function; Rank individuals according to their fitness; Select parents from the population according to some selection procedure; Generate a new population by applying the following operators: crossover, mutation, and reproduction; Compute the fitness of the individuals of the new population; Until (The sopping criteria are satisfied) Repeat from 5
Table-1 records the number of columns (Nbcols) with C1P rounded to the nearest integer

Summary

Introduction

The problem of consecutive ones submatrix on a binary matrix has been known since the 1950s. It was suggested by Fulkerson and Gross [1] and described as follows: Let be an incidence matrix, and can we rearrange its columns so that each row has a single block of ones?. A block of 1's (block of 0's) in a binary matrix A is any maximal set of consecutive one entries (zero entries) appearing in the same row [2]. A binary matrix A is said to have the Consecutive Ones Property (C1P) if the ones in every row appear consecutively [3]. 10s are consecutive in every row [3] This property is similar for the columns, through matrix transposition A binary matrix has the C1P for rows if its columns can be ordered such that all the

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