| Summary: | 博士 === 國立臺灣大學 === 資訊工程學研究所 === 100 === Given a positive integer k and an undirected edge-weighted connected simple graph G with n edges, where k ≤ n, we wish to partition the graph into k edge-disjoint connected components of approximately the same size. We focus on the max-min ratio of the partition, which is the weight of the maximum component divided by that of the minimum component. For k = 2,3, and 4, it has been shown that the upper bound of the max-min ratio of an unweighted tree is two. For any k, the best previous upper bound of the max-min ratio of an unweighted tree is three.
In this thesis, for any graph with no edge of weight larger than one half of the average weight of a component, we provide a linear-time algorithm for delivering a partition with max-min ratio at most two. Together with the fact that the max-min ratio is at least two for some instances, we have that the max-min ratio upper bound attained in this thesis is tight. Furthermore, by an extreme example, we show that the above restriction on edge-weights is the loosest possible.
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