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3-opt

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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 .[1] Iterated 3-opt, in which passes are repeated until no more improvements can be found, has a higher time complexity.

In an array representation of a tour with nodes , where , deleting three edges separates into four segments , and . Note that the segments and are connected. The reconnection of the Subtours is done by a combination of reversing one or both of and and switching their respective positions. The 8 possible new tours are , , , , , ,  and . Here  indicates that segment  is reversed.

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