In general, one method to find good, if not optimal, solutions is based on insects depositing pheromones as they walk; in addition to the ant colony optimization, one can start with a random cycle and trade adjacent pairs.
If the central structure of this problem is Euclidean, then the Christofides algorithm can be applied to it. The Lin–Kernighan heuristic is used on this problem, and solutions can be improved using the 3-opt or 2-opt algorithms, a refinement of which is simulated annealing. In general, one method to find good, if not optimal, solutions is based on insects depositing pheromones as they walk; in addition to the ant colony optimization, one can start with a random cycle and trade adjacent pairs. Name this NP-hard problem of finding a minimum-total-weight Hamiltonian cycle on a weighted graph, analogized to a certain type of person visiting many cities while going the least distance.
If the central structure of this problem is Euclidean, then the Christofides algorithm can be applied to it.
traveling salesman problem [accept shortest tour problem]