網路設施放置理論 (network location theory) 在傳統上的討論都假設網路當中的節點 (vertices) 與邊 (edges) 的權重 (weights) 均為已知。然而，在現實當中，這些值經常包含不確定性 (uncertainty)，且無法精確地估算。在近期當中，出現了越來越多關於最小後悔設施放置問題 (minmax-regret location problems) 的研究。在最小後悔 (minmax-regret) 模型 (model) 當中，各參數分別以一個區間來表示其不確定性，並期望評估函數在最差情況下的差值要越小越好。在傳統問題裏，知名的 p-重心問題 (p-median problem) 在一般圖 (general graphs) 上是 NP-hard，但在樹狀圖 (trees) 上則存在時間複雜度為 O(pn^2) 的演算法。目前為止，當 p >= 2 時，minmax-regret 版本的 p-median problem，即使是在樹狀圖上都沒有存在多項式時間 (polynomial-time) 的演算法。本論文在樹狀圖上討論 minmax-regret 版本的 2-重心問題 (the minmax-regret 2-median problem)。我們假設樹狀圖上每條邊都有一個非負的長度 (length)，而每個節點則有包含不確定性的權重。此問題一個直覺的窮舉做法時間複雜度為 O(n^5)。本論文提出一個比較有效率，時間複雜度為 O(n^4) 的演算法。改進的關鍵是利用了節點之間的平分邊 (bisectors)，其中 bisector 的定義如下：給定兩個節點，此兩節點間之路徑 (path) 上，包含中點 (middle point) 的邊稱為此兩節點的 bisector。

Network location theory was traditionally concerned under the assumption that the vertex weights and edge lengths in the network are known precisely. However, in reality, these values usually involve uncertainty and cannot be estimated accurately. Recently, increasing research efforts have been devoted to minmax-regret location problems, in which uncertainty of network parameters are characterized by intervals and the goal is to minimize the worst case loss in the objective function. The well-known p-median problem is NP-hard on general networks, but admits an efficient O(pn^2)-time algorithm on trees. So far, even on trees, there is no polynomial-time algorithm for the p-median problem on the minmax-regret model with p >= 2. In this thesis, we study the minmax-regret 2-median problem on a tree, in which each edge is associated with a non-negative length and the weight of each vertex is uncertain. A naïve approach to this problem requires O(n^5) time. In this thesis, a more efficient algorithm is presented, which requires O(n^4) time. The main idea of our O(n^4)-time algorithm is to utilize bisectors between vertices, where a bisector is an edge containing the middle point of the path between two vertices.

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