This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. In
Part II, we consider the computational complexity of ORPs arising in genetic
algorithms for problems on permutations: the Travelling Salesman Problem, the
Shortest Hamilton Path Problem and the Makespan Minimization on Single
Machine and some other related problems. The analysis indicates that the
corresponding ORPs are NP-hard, but solvable by faster algorithms, compared
to the problems they are derived from.
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results. Part I presents the basic principles of optimal recombination with a
survey of results on Boolean Linear Programming Problems. Part II (to appear
in a subsequent issue) is devoted to the ORPs for problems which are
naturally formulated in terms of search for an optimal permutation.
We propose a new genetic algorithm with optimal recombination for the asymmetric instances of travelling salesman problem. The algorithm incorporates several new features that contribute to its effectiveness: 1. Optimal recombination problem is solved within crossover operator. 2. A new mutation operator performs a random jump within 3-opt or 4-opt neighborhood. 3. Greedy constructive heuristic of W. Zhang and 3-opt local search heuristic are used to generate the initial population. A computational experiment on TSPLIB instances shows that the proposed algorithm yields competitive results to other well-known memetic algorithms for asymmetric travelling salesman problem.
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