| Summary: | 博士 === 國立清華大學 === 資訊工程學系 === 88 === This dissertation studies the oldest areas of inquiry in graph theory. The hamiltonian path problem, for a given graph, is to find a path to traverse each vertex exactly. For a general graph, it has been shown that this problem is NP-complete and it is widely believed that it will unlikely have any efficient algorithm.
In this dissertation, we shall first try to solve the hamiltonian path problem on series-parallel graph, which is a special class of graphs encountered frequently in circuit design. We shall use the dynamic programming approach to provide a linear time algorithm for this problem.
In the second portion of this dissertation, we discuss the weighted hamiltonian path problem, and try to analyze the complexity of this optimization problem. We shall show that the weighted hamiltonian path problem is NPO-complete, and thus establish the hardness in approximating this problem.
The third portion is the weighted hamiltonian path completion problem, which is a more interesting variation of the weighted hamiltonian path problem. Given any graph, we are required to find an edge set to add into this graph so that it can have a hamiltonian path. We show that this problem is very difficult to approximate. It will unlikely have any
constant ratio approximation algorithm even when the given graph is a tree. Moreover, it still remains NP-hard when the edge weights are restricted to be either 1 or 2. We then propose an approximation algorithm with performance ratio 2,
and prove that this problem has no polynomial-time approximation scheme (PTAS) unless NP=P. Furthermore, we give an
approximation algorithm with performance ratio 1.5 for k-stars.
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