Traveling Salesperson Problem（TSP）是一個相當有名的NP-Completeness問題，在過去有各式各樣的相關演算法被提出來。它所涵蓋的問題相當多，而且許多問題之間會有A問題包含B問題的關係存在。只要有這種關係存在，我們就可以使用A問題的Algorithm去解B問題。例如，以ETSP(Euclidean TSP)、STSP（Symmetric TSP）和ATSP（Asymmetric TSP）這三種著名的TSP問題來說，ETSP的問題包含於STSP的問題，而STSP的問題則包含於ATSP的問題中。若我們為ATSP問題發展出一個求最佳解的Algorithm，則它不但可以解決ATSP的問題，也可以用來解決STSP和ETSP的問題。
就目前所知，用來求ATSP問題最佳解的較好的方法有variable terpoles及LC branch-and-bound的Algorithm。而STSP及ETSP問題則尚無較有效率的解法，一般人在解這兩種問題時，還是會使用為ATSP問題所設計的branch-and-bound algorithm來求。當我們得知這樣的訊息後感到好奇，為什麼ETSP沒有其它更有效率的解法呢？我們是不是可以嘗試去找出一個更好的方法？在本篇論文裡，我們試著去找出這些問題的答案。

在本論文中，我們嘗試各種結合Approximation Algorithm及branch-and-bound algorithm的方法，期望能為ETSP問題找出一個更好且更有效率的Algorithm。我們所採用的方法共分下列兩大類：

1. 將Approximation Algorithm當作Preprocessor解出問題的近似解，然後將該值設定給branch-and-bound的Upper Bound，再跑branch-and-bound程式。

2. 在每些nodes上跑一次Approximation Algorithm得近似解，branch-and-bound程式再根據該值做適當的Bounding動作。

Traveling Salesperson Problem is a very famous problem of NP-Completeness, and all kinds of algorithms were been issued. This problem covers many sub-problems, and the relation that some sub-problems B include other sub-problems A exists. If the relationship exists, we can use the algorithm for solving problem A to find the solution for problem B. For example, Euclidean TSP (ETSP), Symmetric TSP (STSP) and Asymmetric TSP (ATSP) are three noted sub-problems of the TSP. There exists the relation, which ETSP is included in STSP and STSP is included in ATSP among them. In other words, if we found an optimization algorithm for the ATSP, it also can be used to solve STSP and ETSP.
As everyone knows, two good branch-and-bound algorithms for solving ATSP are variable terpoles and LC branch-and-bound algorithms, but no any good branch-and-bound algorithms for the STSP and ETSP presently exists. We are very curious that why doesn’t the algorithm for the ETSP exist and can we find out a good branch-and-bound algorithm for it.

We tried all kinds of ways, which combine the approximation algorithm and branch-and-bound algorithm in my thesis, and we attempted to find the branch-and-bound algorithms, which are better and more efficient than original ones. There are two classifications in our methods:

1. The approximation algorithm is used as the preprocessor of the branch-and-bound algorithm.

2. To run approximation algorithm once at each node for getting the approximate solution, then to bound the appropriate nodes according to the solution.

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