我們研究兩個最小總和排程問題，天際線排程問題與智慧電網排程問題
首先，我們考慮在使用較高顏色索引時成本隨之增加的環境下進行線段著色問題。線上任何點的顏色成本是該點使用的最大顏色索引的成本，而整體著色的成本是數線上所有點成本的加總。最低成本的著色稱為最低天際線著色問題。我們證明了計算最小天際線著色問題是很難的，並開始研究線段逐一到達的線上設定。對於線性顏色成本的情況，我們給出了漸近最佳的線上演算法，並給出了該問題的一些延伸的結果。此外，我們考慮了該問題的另一種變化形，其中線段事先分群好且每個要著不同顏色，以最大程度地減少著色成本。我們證明了該變化形問題是很難的，並為此提出了一個兩倍近似演算法。
此外，我們研究了單位大小輸入的智慧電網問題中的線上排程。我們證明貪婪法是解決該問題的最佳線上演算法。通常，要證明是最佳的線上演算法是通過指出競爭率的上限並顯示具有相對應競爭率的下限。但是，我們的分析沒有採用這種方法。相反地，我們直接證明了最佳性而沒有給出競爭比率的確切界限。這個結果直接影響到指定機器安排問題的結果。

We study two min-sum scheduling problems, skyline scheduling problem and smart grid scheduling problem.
First, we consider the interval coloring problem in a setting where an increasing cost is associated with using a higher color index. The cost of a coloring at any point of the line is the cost of the maximum color index used at that point, and the cost of the overall coloring is the integral of the cost over all points on the line. A coloring of minimum cost is called a minimum skyline coloring. We prove that the problem of computing a minimum skyline coloring is NP-hard and initiate the study of the online setting, where intervals arrive one by one. We give an asymptocially optimal online algorithm for the case of linear color costs and present further results for some generalizations of the problem. Also, we consider the variant of the problem where the intervals are grouped in advance and each is colored by a distinct color to minimize the cost of the coloring. We show that this problem variant is NP-hard and present a 2-approximation algorithm for it.
In addition, we study online scheduling of unit-sized jobs in the smart grid problem. We show that the greedy algorithm is an optimal online algorithm for the problem. Typically, an online algorithm is proved to be an optimal online algorithm through bounding its competitive ratio and show- ing a lower bound with matching competitive ratio. However, our analysis does not take this approach. Instead, we prove the optimality without giving the exact bounds on competitive ratio. The results have direct implication on restricted assignment problem.

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