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Evolution of new algorithms for the binary knapsack problem


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.

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Binary knapsack problem


Genetic programming


Genetic programming platform for evolving tree structures of code


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


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


Uncorrelated instance of the binary knapsack problem


Weakly correlated instance of the binary knapsack problem


Strongly correlated instance of the binary knapsack problem


Subset sum instance of the binary knapsack problem


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


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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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Correspondence to Lucas Parada.

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Parada, L., Herrera, C., Sepúlveda, M. et al. Evolution of new algorithms for the binary knapsack problem. Nat Comput 15, 181–193 (2016).

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