圖的均勻邊切割問題

Abstract

當給定正整數 k 以及對於一個邊數為 n 的無向連通圖，我們關心的是把此圖的邊分割成 k 個連通子圖，使得每一部份的大小盡可能地均勻。在此篇論文以最大塊與最小塊的邊數比來當作衡量的標準。在此課題上，過去的文獻只探討當圖限制為樹的結構，且邊為無權重之情況。對於 k=2,3 及 4 的情況，已被證明最大塊與最小塊的邊數比可以保證在 2 以內。而對於任意的 k ，先前最佳的結果是保證最大塊與最小塊的邊數比在 3 以內。 此篇論文將圖推廣至一個邊有權重的任意連通圖。當每個邊的權重不超過平均權重的一半時，我們提出一個線性時間的演算法來得到此圖的邊切割，使得最大塊與最小塊的邊數比可以保持在 2 以內。此外，我們也藉由一個例證來說明邊權重的限制是最寬鬆的。

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.