3-opt
Introduction
The 3-opt algorithm is a method used in the field of operational research and computer science to solve the Traveling Salesman Problem (TSP) and related problems. It is a local search method used for solving combinatorial optimization problems, specifically in the field of heuristics.
Overview
The 3-opt algorithm is a specific case of the k-opt algorithm, where k equals 3. The k-opt algorithm is a method for solving the TSP and similar problems by iteratively removing 'k' edges from a graph and reconnecting them in a different way to produce a new and shorter tour. The 3-opt method specifically involves removing three edges and reconnecting the remaining nodes in one of the seven possible ways that do not result in a disconnected tour.
Methodology
The 3-opt algorithm starts with an initial solution to the TSP, which could be a random tour or a tour generated by some heuristic. The algorithm then examines all possible combinations of three edges in the tour. For each combination, it checks if reconnecting the nodes in a different way would result in a shorter tour. If such a reconnection is found, the algorithm replaces the current tour with the new, shorter tour, and starts the process again.
The 3-opt algorithm continues this process until it can no longer find any three edges that, when reconnected, produce a shorter tour. At this point, the algorithm terminates and returns the current tour as the solution.
Efficiency and Performance
The 3-opt algorithm is more computationally intensive than the 2-opt algorithm, as it needs to examine a larger number of combinations of edges. However, it has the potential to find better solutions, as it can escape from local optima that the 2-opt algorithm might get stuck in.
In terms of time complexity, the 3-opt algorithm is O(n^3), where 'n' is the number of nodes in the graph. This is because, in the worst case, the algorithm needs to examine all possible combinations of three edges, which is proportional to the cube of the number of nodes.
Despite its higher time complexity, the 3-opt algorithm is often used in practice, especially in combination with other heuristics, to solve the TSP and similar problems. It is particularly effective when used as part of a multi-start or iterated local search strategy, where the algorithm is run multiple times from different initial solutions.
Variations and Extensions
There are several variations and extensions of the 3-opt algorithm. One common variation is the variable-opt method, where the value of 'k' is not fixed but varies during the search. This method starts with a small value of 'k' and increases it if no improvement can be found, allowing the search to escape from local optima.
Another extension is the guided 3-opt method, which uses additional information to guide the search. This information could be, for example, a prediction of the likelihood that a certain reconnection will lead to an improvement, based on previous search steps.