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Natural Computing

, Volume 15, Issue 1, pp 181–193 | Cite as

Evolution of new algorithms for the binary knapsack problem

  • Lucas Parada
  • Carlos Herrera
  • Mauricio Sepúlveda
  • Víctor Parada
Article

Abstract

Due to its NP-hard nature, it is still difficult to find an optimal solution for instances of the binary knapsack problem as small as 100 variables. In this paper, we developed a three-level hyper-heuristic framework to generate algorithms for the problem. From elementary components and multiple sets of problem instances, algorithms are generated. The best algorithms are selected to go through a second step process, where they are evaluated with problem instances that differ in size and difficulty. The problem instances are generated according to methods that are found in the literature. In all of the larger problem instances, the generated algorithms have less than 1 % error with respect to the optimal solution. Additionally, generated algorithms are efficient, taking on average fractions of a second to find a solution for any instance, with a standard deviation of 1 s. In terms of structure, hyper-heuristic algorithms are compact in size compared with those in the literature, allowing an in-depth analysis of their structure and their presentation to the scientific world.

Keywords

Automatic generation of algorithms Combinatorial optimization Evolutionary computation Genetic programming Hyper-heuristic Knapsack problem 

Abbreviations

BKP

Binary knapsack problem

GP

Genetic programming

GPC++

Genetic programming platform for evolving tree structures of code

IKL

In the knapsack list; data structure used by the evolved algorithms

OKL

Out of knapsack list; data structure used by the evolved algorithms

UC

Uncorrelated instance of the binary knapsack problem

WC

Weakly correlated instance of the binary knapsack problem

SC

Strongly correlated instance of the binary knapsack problem

SS

Subset sum instance of the binary knapsack problem

FC

Fitness cases i.e. problem instances of the binary knapsack problem used to evolve algorithms

Notes

Acknowledgments

The authors would like to thank the Complex Engineering Systems Institute ICM: P-05-004-F, CONICYT: FBO16, DICYT: 61219-USACH, ECOS/CONICYT: C13E04, STICAMSUD: 13STIC-05.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Lucas Parada
    • 1
  • Carlos Herrera
    • 1
  • Mauricio Sepúlveda
    • 2
  • Víctor Parada
    • 2
  1. 1.Departamento de Ingeniería IndustrialUniversidad de ConcepciónConcepciónChile
  2. 2.Departamento de Ingeniería InformáticaUniversidad de Santiago de ChileSantiagoChile

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