In this paper, we present a comprehensive review of the analysis of the well-known $1 - 1/e$ upper bound on the competitiveness that any online algorithm can achieve, as established in the classical paper by Karp, Vazirani, and Vazirani (STOC 1990). We discuss in detail all the minor and major technical issues in their approach and present a \emph{simple yet rigorous} method to address them. Specifically, we show that the upper bound of $n(1 - 1/e) + o(n)$ on the performance of any online algorithm, as shown in the paper, can be replaced by $\lceil n \cdot (1 - 1/e) + 2 - 1/e \rceil$. Our approach is notable for its simplicity and is significantly less technically involved than existing ones.

翻译：本文对Karp、Vazirani与Vazirani（STOC 1990）经典论文中建立的关于任何在线算法可达到的竞争性上界$1 - 1/e$的分析进行了全面回顾。我们详细讨论了其方法中所有次要和主要的技术问题，并提出了一种\emph{简洁而严谨}的方法来解决这些问题。具体而言，我们证明原论文中关于任何在线算法性能的上界$n(1 - 1/e) + o(n)$可替换为$\lceil n \cdot (1 - 1/e) + 2 - 1/e \rceil$。我们的方法以其简洁性著称，且在技术复杂性上显著低于现有方法。