A double-loop hybrid algorithm for the traveling salesman problem with arbitrary neighbourhoods

This paper addresses the traveling salesman problem with arbitrary neighbourhoods, which is an NP-hard problem in combinatorial optimization and is important in operations research and theoretical computer science. Existing methods based on neighbourhood predigestion and discretization are impractical in cases in which the neighbourhoods are arbitrary. In this paper, the neighbourhoods of the traveling salesman problem are generalized to arbitrarily connected regions in planar Euclidean space. A novel approach to solving this problem is proposed, including a boundary-based encoding scheme and a double-loop hybrid algorithm based on particle swarm optimization and genetic algorithm. In the hybrid algorithm, linear descending inertia weight particle swarm optimization is adopted to search continuous visiting positions in the outer loop, and the genetic algorithm is used to optimize the discrete visiting sequence in the inner loop. The boundary-based encoding scheme can reduce the search space significantly without degrading the solution quality, and the hybrid algorithm can find a high-quality solution in a reasonable time. The computational results on both small and large instances demonstrate that the proposed approach can guarantee a high-quality solution in a reasonable time, compared with three other state-of-the-art algorithms: iterative algorithm, branch-and-bound algorithm, and upper and lower bound algorithm. Moreover, the proposed approach works efficiently in a real-world application that cannot be solved by existing algorithms.