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Unformatted text preview: 2.12 Application: Exponential Neighbourhood Local Search for the TSP 83 is normally far from optimal but is much better than a random solution 7 . (Some construction heuristics for the TSP are described later in this book.) In the second phase, a local search heuristic is used. During every iteration of this heuristic, a neighbourhood of a current best solution is considered and a better solution (in certain cases, the best solution in the neighbourhood) is found. When no better solution in the neighbourhood exists the heuristic terminates. (There are several variations of the above description .) In many cases, so-called 3-Opt is applied. In 3-Opt, the neighbourhood of a hamiltonian cycle C consists of all tours in ↔ K n obtained from C by deleting three arcs and then adding three arcs. (This notion can be easily generalized to k-Opt for every fixed k ≥ 3.) The cardinality of this neighbourhood is certainly Θ ( n 3 ). Also, O ( n 3 ) time is required to completely search this neigh- bourhood (i.e., to find the best hamiltonian cycle) if we look at the tours of the neighbourhood one by one. Certainly, the cubic time is unacceptable for large-scale instances of the TSP. However, 3-Opt is widely used in practice since usually only a small fraction of the neighbourhood is searched before a better solution is found. Despite the fact that 3-Opt allows one to find quite good solutions to large-scale instances of the TSP, the way of looking at the solutions one by one seems rather inefficient. Therefore, in the 1980’s, Sarvanov and Doroshko [651, 652] and Gutin  introduced independently some neighbourhoods of exponential size where the best solution can be obtained in polynomial time. Recently, var- ious neighbourhoods of exponential size for the TSP were suggested and investigated (see, e.g., Balas and Simonetti , Burkard, Deineko and Woeg- inger , Glover , Glover and Punnen , Potts and Velde  and Punnen ). The paper  by Deineko and Woeginger is an excellent survey on the topic. Balas and Simonetti  and Carlier and Villon  constructed and implemented local search algorithms which use exponential neighbourhoods. Their results are very encouraging. They also show the ne- cessity of further theoretical research on the topic. There are different types of neighbourhoods for the TSP; many of them can be found in [188, 466]. The following definition of a neighbourhood struc- ture for the TSP is due to Deineko and Woeginger . In this definition, we assume that every tour T = π (1) π (2) ...π ( n ) π (1) starts from the ver- tex 1, i.e., π (1) = 1. Therefore, we will identify T with the permutation π (1) π (2) ...π ( n ) . A neighbourhood structure consists of a neighbourhood N ( T ) for every tour T such that the neighbourhood N ( π (1) π (2) ...π ( n )) = π * N (12 ...n ) , where π (1) = 1 and * stands for the permutation product (ap- plied from right to left). This definition is somewhat restrictive (for example,plied from right to left)....
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- Spring '11
- Algorithms, Shortest path problem, Digraph, vertices, Digraphs