「設施放置問題」不論就理論研究或實際應用而言，都具有相當的重要性，因此一直受到各個領域的廣泛重視與討論。「設施放置問題」問題討論的是在一個代表網路的圖形上，如何找出一些新設施之最佳設置地點。根據問題的特性，設施的形狀可以是點狀、條狀或是樹狀。在傳統的設施放置問題中，一般會假設節點的權重及邊的長度這些參數都可以被精確的量測。然而實際生活中所蒐集到的資料通常有許多的不確定性，資料不僅會有誤差，並且也可能隨時間產生變化，因此在設施放置問題上考慮不確定性的研究開始受到重視。「最小後悔 (minmax regret)」是用來描述網路之不確定性的重要模型之一。
這篇論文研究的主題是在「最小後悔」這個模型上討論條狀設施放置問題中最重要的三個問題：path median、path centdian 和 path center。由於這三個問題在 general graph 上是 NP-hard，我們將重點放在討論 tree 的情況。在網路上節點權重有不確定性存在時的情況下，我們將針對這三個問題分別提出比既有方法更快速的演算法。就 minmax regret path median 問題而言，我們提出的演算法將時間複雜度由原本的 O(n4) 降到 O(n2)。就 minmax regret path centdian 問題而言，我們發現的之前的結果有誤，並且提出修正的方法;此外，我們更進一步提出改進演算法，將時間複雜度由原本的 O(n5 log n) 降到 O(n4)。就 minmax regret path center 問題而言，我們將原本的 O(n2) 時間的演算法改善到 O(n log n)。

In the fields of communication and transportation, many researchers have been concentrating on location problems. The objective of a location problem is to optimally choose the location of facilities. In network location theory, the shapes of facilities can be points, paths, or trees. A network usually involves two types of parameters: weights of nodes and lengths of edges. Traditionally, the node weights and edge lengths of a network are assumed to be known precisely. However, the weights and lengths of a network may fluctuate or be inaccurate due to poor measurements. Thus, location models involving uncertainty have attracted increasing research efforts in recent years. One of the most important ways for modeling network uncertainty is the minmax regret approach.
In this dissertation, we examine the path center, median and centdian problems on the minmax-regret model. Since the three problems on general graphs are NP-hard, we focus on the problems on trees. We present efficient algorithms for the minmax regret path median, path centdian, and path center problems on a tree with uncertain vertex weights. For the minmax regret path median problem, we improve the upper bound from O(n4) to O(n2). For the minmax regret path centdian problem, we show that there is a small bug in the previous solution and show how to rectify the bug; in addition, we further improve the upper bound from O(n5 log n) to O(n4). For the minmax regret path center problem, we improve the previous upper bound from O(n2) to O(n log n).

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