3-opt

In optimization, 3-opt is a simple local search heuristic for finding approximate solutions to the travelling salesperson problem and related network optimization problems. Compared to the simpler 2-opt algorithm, it is slower but can generate higher-quality solutions.

3-opt analysis involves deleting three edges from the current solution to the problem, creating three sub-tours. There are eight ways of connecting these sub-tours back into a single tour, one of which consists of the three deleted edges. These reconnections are analysed to find the optimum one. This process is then repeated for a different set of 3 connections, until all possible combinations have been tried in a network. A single pass through all triples of edges has a time complexity of $$O(n^3)$$. Iterated 3-opt, in which passes are repeated until no more improvements can be found, has a higher time complexity.