<Li> Convert to TSP using shortcuts . </Li> <P> The pairwise exchange or 2 - opt technique involves iteratively removing two edges and replacing these with two different edges that reconnect the fragments created by edge removal into a new and shorter tour . Similarly, the 3 - opt technique removes 3 edges and reconnects them to form a shorter tour . These are special cases of the k - opt method . Note that the label Lin--Kernighan is an often heard misnomer for 2 - opt . Lin--Kernighan is actually the more general k - opt method . </P> <P> For Euclidean instances, 2 - opt heuristics give on average solutions that are about 5% better than Christofides' algorithm . If we start with an initial solution made with a greedy algorithm, the average number of moves greatly decreases again and is O (n) (\ displaystyle O (n)). For random starts however, the average number of moves is O (n log ⁡ (n)) (\ displaystyle O (n \ log (n))). However whilst in order this is a small increase in size, the initial number of moves for small problems is 10 times as big for a random start compared to one made from a greedy heuristic . This is because such 2 - opt heuristics exploit ` bad' parts of a solution such as crossings . These types of heuristics are often used within Vehicle routing problem heuristics to reoptimize route solutions . </P> <P> Take a given tour and delete k mutually disjoint edges . Reassemble the remaining fragments into a tour, leaving no disjoint subtours (that is, don't connect a fragment's endpoints together). This in effect simplifies the TSP under consideration into a much simpler problem . Each fragment endpoint can be connected to 2k − 2 other possibilities: of 2k total fragment endpoints available, the two endpoints of the fragment under consideration are disallowed . Such a constrained 2k - city TSP can then be solved with brute force methods to find the least - cost recombination of the original fragments . The k - opt technique is a special case of the V - opt or variable - opt technique . The most popular of the k - opt methods are 3 - opt, and these were introduced by Shen Lin of Bell Labs in 1965 . There is a special case of 3 - opt where the edges are not disjoint (two of the edges are adjacent to one another). In practice, it is often possible to achieve substantial improvement over 2 - opt without the combinatorial cost of the general 3 - opt by restricting the 3 - changes to this special subset where two of the removed edges are adjacent . This so - called two - and - a-half - opt typically falls roughly midway between 2 - opt and 3 - opt, both in terms of the quality of tours achieved and the time required to achieve those tours . </P>

How big is the state space of this tsp problem