# A Comparison of Binary and Integer Encodings in Genetic Algorithms for the Maximum k-Coverage Problem with Various Genetic Operators

## TL;DR

This paper compares binary and integer encoding in genetic algorithms for solving the maximum k-coverage problem, showing how different crossover and repair methods affect performance.

## Contribution

The study introduces and evaluates novel combinations of crossover and repair operators for binary and integer encodings in genetic algorithms for the MKCP.

## Key findings

- Binary encoding with uniform crossover and random repair improved average objective values by up to 3.24%.
- Integer encoding with uniform crossover and conservative repair improved average performance by up to 4.47%.
- Conservative repair was more effective with uniform crossover than with one-point crossover in integer encoding.

## Abstract

The maximum k-coverage problem (MKCP) is a problem of finding a solution that includes the maximum number of covered rows by selecting k columns from an m ×
n matrix of 0s and 1s. This is an NP-hard problem that is difficult to solve in a realistic time; therefore, it cannot be solved with a general deterministic algorithm. In this study, genetic algorithms (GAs), an evolutionary arithmetic technique, were used to solve the MKCP. Genetic algorithms (GAs) are meta-heuristic algorithms that create an initial solution group, select two parent solutions from the solution group, apply crossover and repair operations, and replace the generated offspring with the previous parent solution to move to the next generation. Here, to solve the MKCP with binary and integer encoding, genetic algorithms were designed with various crossover and repair operators, and the results of the proposed algorithms were demonstrated using benchmark data from the OR-library. The performances of the GAs with various crossover and repair operators were also compared for each encoding type through experiments. In binary encoding, the combination of uniform crossover and random repair improved the average objective value by up to 3.24% compared to one-point crossover and random repair across the tested instances. The conservative repair method was not suitable for binary encoding compared to the random repair method. In contrast, in integer encoding, the combination of uniform crossover and conservative repair achieved up to 4.47% better average performance than one-point crossover and conservative repair. The conservative repair method was less suitable with one-point crossover operators than the random repair method, but, with uniform crossover, was better.